Optimizing joint location-scale monitoring – An adaptive distribution-free approach with minimal loss of information

Accepted Manuscript

Optimizing joint location-scale monitoring - an adaptive distribution-free approach with minimal loss of information Zhi Song, Amitava Mukherjee, Yanchun Liu, Jiujun Zhang PII: DOI: Reference:

S0377-2217(18)31001-4 https://doi.org/10.1016/j.ejor.2018.11.060 EOR 15509

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

4 April 2018 22 November 2018 24 November 2018

Please cite this article as: Zhi Song, Amitava Mukherjee, Yanchun Liu, Jiujun Zhang, Optimizing joint location-scale monitoring - an adaptive distribution-free approach with minimal loss of information, European Journal of Operational Research (2018), doi: https://doi.org/10.1016/j.ejor.2018.11.060

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Highlights • An adaptive nonparametric Shewhart-Lepage scheme for joint monitoring is developed. • A new simplified adaptive scheme is proposed based on small sample behavior . • The adaptive schemes use information about tail-weight of the process distribution.

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• On the whole the modified adaptive scheme is superior for various distributions.

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Authors:

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Optimizing joint location-scale monitoring - an adaptive distribution-free approach with minimal loss of information

Zhi Song1,2 , Amitava Mukherjee3 , Yanchun Liu4 , Jiujun Zhang2,∗ Affiliations:

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1. College of Science, Shenyang Agricultural University, Shenyang 110866, P.R.China. 2. Department of Mathematics, Liaoning University, Shenyang 110036, P.R.China.

3. Production, Operations and Decision Sciences Area XLRI-Xavier School of Management, XLRI Jamshedpur, Jamshedpur 831001, India.

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4. Business School, Liaoning University, Shenyang 110036, P.R.China.

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E-mail addresses:

[email protected](Z. Song)

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[email protected], [email protected](A. Mukherjee)

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[email protected](Y. Liu) [email protected](J. Zhang)

* Corresponding author.

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Optimizing joint location-scale monitoring - an adaptive distribution-free

Abstract

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approach with minimal loss of information

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Nonparametric statistical process monitoring (NSPM) schemes are very effective in monitoring various non-normal and complex processes. Nevertheless, there is one disadvantage of the traditional NSPM schemes. We often lose some information during scoring or ranking. For example, we sometimes ignore the information related to the shape or tail-weights of the underlying process distribution. In this paper, we introduce distribution-free adaptive Shewhart-Lepage (SL) type schemes for simultaneous monitoring of location and scale parameters using information about symmetry and tail-weights of the process distribution. We consider an adaptive SL type scheme, referred to as the LPA scheme, based on the three modified Lepage-type statistics. Using numerical results obtained via Monte-Carlo, and considering operational simplicity, we also propose a new adaptive SL type scheme, referred to as the MLPA scheme, with finite sample correction. These adaptive approaches use the Phase-I data to assess the tail-weights of the process distribution, and then select an appropriate SL type statistic for process monitoring. Consequently, these schemes correct the disadvantage of the traditional distribution-free schemes to a great extent. We compare the two adaptive schemes with the three individual SL type schemes, and also with the classical SL and Shewhart-Cucconi (SC) schemes in terms of the average, the standard deviation and some percentiles of the run length distribution. Numerical results establish that the MLPA scheme is superior for jointly monitoring the parameters of a broad class of process distributions belong to the location-scale family.

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Keywords: Quality control; Adaptive Shewhart-Lepage (SL) type scheme; Nonparametric; Simultaneous monitoring; Average run length

Introduction

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In quality control and management, statistical process monitoring (SPM) plays a transformative role. From

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quality monitoring in manufacturing industries to health care surveillance, from monitoring service quality to network traffic, the SPM schemes are used as a fundamental tool. Several SPM schemes are introduced in the past nine decades assuming that the underlying process distribution is known. For example, traditional ¯ or R schemes are based on the assumption that the process distribution is normal. Similarly, the Shewhart X p-chart assumes that the process distribution is binomial. Such SPM schemes are referred to as parametric monitoring schemes. For the implementation of these schemes, we require the complete knowledge of the underlying process distribution. Readers may see Montgomery (2009) or Qiu (2014) for more details. For more complicated schemes based on parametric procedures and for recent developments, we recommend the works by Haridy, Wu, Lee, and Bhuiyan (2013), Yeong, Khoo, Lee, and Rahim (2013), Liu, Yu, Ma, and Tu (2013), Lee (2013), Nenes and Panagiotidou (2013), Bersimis, Koutras, and Maravelakis (2014), Leoni, 1

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Costa, and Machado (2015), Nikolaidis and Tagaras(2017), Teoh et al. (2017) and Goedhart et al. (2017). In various applications, most of the data streams follow complex processes and their exact distributions are often untraceable. When a parametric distribution specified beforehand is invalid, some authors, such as Qiu and Zhang (2015), Koutras and Triantafyllou (2018), among others, argue that results from such conventional parametric monitoring schemes would no longer be reliable. To address this problem, the nonparametric approach is often considered, because, in these situations, a nonparametric monitoring scheme is generally more robust and effective than a parametric one. For a thorough account of the nonparametric statistical process monitoring (NSPM) schemes, see, Bakir (2006), Qiu and Li (2011a, 2011b), Mukherjee, Graham,

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and Chakraborti (2013) and Chakraborti, Qiu, and Mukherjee (2015), He, Jiang, and Deng (2018) and references therein. Interested readers may go through the chapters 8 and 9 of the book by Qiu (2014) for further details on the NSPM schemes.

In practice, shifts in both the location and scale parameters of a process may happen simultaneously. Therefore, nonparametric schemes based on a single plotting statistic for jointly monitoring the location and scale parameters have received a great deal of attention in recent years. Among various NSPM schemes

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for joint monitoring, the well-known Lepage-type statistics have gained more attention. Mukherjee and Chakraborti (2012) first proposed a distribution-free Shewhart-Lepage (SL) scheme for jointly monitoring the location and scale parameters of a process. Chowdhury, Mukherjee, and Chakraborti (2015) further considered a cumulative sum scheme based on the Lepage statistic. Oprime, Toledo, Gonzalez, and Chakraborti (2016) presented a regression based method for determining the control limits of the SL scheme. Chong, Mukherjee, and Khoo (2017) proposed a fuzzy monitoring scheme that combines the merits of both the SL type distance and SL type max schemes, and the same authors (2018) considered a class of SL type schemes

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for one-sided joint monitoring. Mukherjee (2017) introduced some distribution-free exponentially weighted moving average schemes based on the Lepage statistic. Mahmood, Nazir, Abbas, Riaz, and Ali (2017)

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carried out an in-depth performance evaluation of the SL and Shewhart-Cucconi (SC) monitoring schemes. Mukherjee and Marozzi (2017a) introduced a circular-grid version based on some Lepage-type statistics. Mukherjee and Sen (2018) considered a class of percentile modified SL type schemes. For more contribu-

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tions on nonparametric joint monitoring schemes, one may see, for example, Chowdhury, Mukherjee, and Chakraborti (2014), Celano, Castagliola, and Chakraborti (2016), Li, Xie, and Zhou (2016), Mukherjee and Marozzi (2017b) and Zhang, Li, and Li (2017).

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In industrial process control, nonparametric techniques are slowly gaining popularity, though not as fast as expected. Woodall and Montgomery (2014) noted that the lack of availability of software may be the reason. Moreover, most nonparametric methods seem to sacrifice too much of the basic information

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in the sample. For example, when an underlying process is indeed normally distributed, a nonparametric method may be slightly less efficient compared to the parametric schemes based on sample mean and variance. However, nonparametric schemes can be much more efficient when the normality assumption is invalid. Qiu (2018) outlined some perspectives on different issues related to the strengths and limitations of various nonparametric monitoring schemes. He pointed out that “the loss of information is just the price to pay for the nonparametric control charts to be robust without specifying a parametric form for describing the in-control process distribution. One important future research topic is to minimize the lost information while keeping the favorable properties of the nonparametric control charts.” Hogg (1974) coined the idea of adaptive tests that minimize the loss of information in nonparametric testing of hypothesis. In adaptive inference, we first use the available data to estimate the tail-weight and skewness of the underlying 2

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distribution. Thereafter, an appropriate test is selected for the classified type of distribution. Clearly, an adaptive test does not completely overlook the information about the tail-weights or the degree of asymmetry in the original data. Thus adaptive approaches fairly eliminate one disadvantage of the nonparametric methods. In this paper, we first introduce a class of distribution-free monitoring schemes based on the modified Lepage-type statistics. Further, motivated by the notion of adaptive inference, we propose two adaptive SL type schemes. One scheme is based on an adaptive test proposed by K¨ossler (2006), and the other is a new modification for finite sample. These adaptive monitoring schemes may not have optimal distributions.

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performance for a particular process distribution, but their performance is near optimal for a wide range of The rest of the paper is organized as follows. In Section 2, we describe a class of Lepage-type statistics and two adaptive Lepage-type statistics. We also discuss the implementation procedures of the proposed class of SL type monitoring schemes. Section 3 is devoted to a detailed performance analysis. We illustrate our proposed monitoring schemes with two realistic datasets from Montgomery (2009) in Section 4. We

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offer a summary and some concluding remarks in Section 5.

Adaptive Lepage-type monitoring schemes

Consider a stable process that is unaffected by any assignable or special causes of variation. In connection to SPM, such a process is called in-control (IC) process. An IC process is subjected to the random variation only. In the present article, we assume the availability of a reference sample of size m from the IC process. To be precise, we consider that a reference sample Xm = (X1 , X2 , . . . , Xm ) is collected from an IC process 0

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with a continuous cumulative distribution function (cdf) F (x). We also assume that Xi s are independently and identically distributed (i.i.d) random variable each having cdf F . Establishment of reference sample

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itself is an interesting research topic. This aspect is covered under Phase-I analysis and is addressed by several researchers in recent times. Interested readers may see Jones-Farmer, Woodall, Steiner, and Champ (2014), Capizzi and Masarotto (2018) for more details. In Phase-I monitoring, practitioners perform a

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retrospective analysis over the historical samples to evaluate the process stability, to identify outliers and to eliminate them. The remaining sample helps practitioners to set up a certain benchmark and is often used as a reference sample for comparing an incoming sequence of test samples in course of a Phase-II analysis.

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We focus only on Phase-II analysis in the present context. Note that, each test sample may consist of a few random observations. In the current paper, we consider a test sample of size n at each stage of Phase-II monitoring. Let Yj = (Yj1 , Yj2 , . . . , Yjn ), j = 1, 2, . . . be the j-th test sample from a cdf G(x). A test 0

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sample is mutually independent of the reference sample. We further assume that Yji s are i.i.d for every i, (1 ≤ i ≤ n) and j, (j ≥ 1). Ideally, two distribution functions F and G should be identical in all respect

when the process is in IC. Nevertheless, when there is a shift in process location or in scale or in both at Phase-II, we often see that G(x) = F ( x−θ δ ), θ ∈ <, δ > 0, where the constants θ and δ represent the

unknown shift in the location and scale parameters, respectively. In classical statistical literature, such a ˇ ak, and Sen (1999). framework is referred to as a general location-scale model. See, for example, H´ajek, Sid´ When θ = 0 and δ = 1, we observe an IC set-up. Note that θ 6= 0 but δ = 1, represents a pure location shift; while θ = 0 and δ 6= 1 indicates a pure scale shift. Finally, if both θ 6= 0 and δ 6= 1, we observe a shift

in both the location and scale parameters.

From the conventional statistical theory of rank tests, the optimal score function corresponding to the 3

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asymptotically most powerful rank test for detecting a pure location shift only in the cdf, say G1 , is given by


0 −g1 G−1 1 (u)
, ϕ1 (u, g1 ) = g1 G−1 1 (u)

0 < u < 1. 0

Here g1 is the probability density function (pdf) of G1 and g1 denotes the first derivative of g1 , where it ˇ ak, and Sen (1999). exists and it is defined to be zero, otherwise, see, for example, H´ajek, Sid´ Similarly, the optimal score function corresponding to the asymptotically most powerful test for detecting

0

ϕ2 (u, g2 ) = −1 −

g2 G−1 2 (u) g2

0


G−1 2 (u)
, G−1 2 (u)

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a pure scale shift in the cdf, say G2 , is given by 0 < u < 1,

where g2 is the pdf of G2 and g2 denotes the first derivative of g2 , where it exists and it is defined to be zero, otherwise. Monitoring statistics are designed under the following three assumptions:

a(k) = ϕ1



k m+n+1

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1. The scores a(k) and b(k) are generated by setting 

and

b(k) = ϕ2



k m+n+1



.

2. The Fisher-information IL (g1 ) and IS (g2 ) are finite, i.e. 1

0

ϕ21 (u, g1 )du

<∞

and

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IL (g1 ) =

Z

IS (g2 ) =

Z

0

1

ϕ22 (u, g2 )du < ∞.

3. The two scores-generating functions ϕ1 (u, g1 ) and ϕ2 (u, g2 ) are orthogonal in the Hilbert space of

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square integrable functions.

We shall now discuss the construction of a class of Lepage-type statistics. Note that the Lepage-type

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statistics are combinations of two statistics, one is an optimal statistic for detecting a shift in location parameter and is based on the score function a(k) while the other one is an optimal statistic for detecting

2.1

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a shift in scale parameter and is based on the score function b(k).

Traditional Lepage-type schemes

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Define the statistics T1 and T2 as T1 =

n X

a(Ri )

n X

b(Ri ).

i=1

and

T2 =

i=1

Here Ri (i = 1, . . . , n) are the ranks of the test sample Yji in the combined sample of size N = m + n during the j-th stage of monitoring. The statistics T1 and T2 are two rank-based statistics, suitable for monitoring a shift in location parameter and in scale parameter respectively. Several researchers studied the properties ˇ ak, and Sen of T1 and T2 and referred to them as the linear rank statistics, see, for example, H´ajek, Sid´ 4

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(1999). When the process is IC, we get E(T1 |IC) =

V ar(T1 |IC) = with a ¯ = (1/N )

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k=1 a(k),

N N n X n X a(k), E(T2 |IC) = b(k) N N k=1

and

k=1

N N X X
2 mn mn b(k) − ¯b (a(k) − a ¯)2 , V ar(T2 |IC) = N (N − 1) N (N − 1) k=1

¯b = (1/N )

k=1

PN

k=1 b(k).

The Lepage-type statistic is the squared Euclidian distance of (0, 0) and given by

T =



T1 − E(T1 |IC) p V ar(T1 |IC)

2

+



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From Randles and Hogg (1971), one can easily see that the statistics T1 and T2 are linearly   uncorrelated. T1 −E(T1 |IC) T2 −E(T2 |IC) √ ,√ V ar(T1 |IC)

T2 − E(T2 |IC) p V ar(T2 |IC)

2

V ar(T2 |IC)

.

from the origin

(1)

Under an IC set-up, T has asymptotically a central χ2 -distribution with 2 degrees of freedom. Further note

V ar(T1 |IC)

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that  uncorrelated, T can also be interpreted as the squared Mahalanobis distance  as T1 and T2 are linearly T1 −E(T1 |IC) T2 −E(T2 |IC) ,√ from origin. of √ V ar(T2 |IC)

The Lepage statistic

The traditional Lepage statistic (LPWAB) is the sum of squares of the standardized Wilcoxon rank sum

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(WRS) statistic for location and the standardized Ansari-Bradley (AB) test for scale. The WRS statisitc and AB statistic are defined by

TLP W AB,1 = WRS =

Ri ,

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i=1

TLP W AB,2 = AB =

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n X

n X Ri − N + 1 . 2 i=1

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It is well known (see Mukherjee and Chakraborti (2012)) that

1 E(TLP W AB,1 |IC) = n(N + 1), 2

V ar(TLP W AB,1 |IC) =

Further,

E(TLP W AB,2 |IC) =

V ar(TLP W AB,2 |IC) =

1 mn(N + 1). 12

  n(N 2 −1) 4N  nN 4

when N is odd when N is even,

  mn(N +1)(N 2 +3) 48N 2 2 −4) mn(N  48(N −1)

when N is odd when N is even.

Clearly, we obtain the original Lepage statistic when T1 is same as the WRS and T2 is same as the AB statistic. For the two-sample location-scale problem, the Lepage test has received more attention, since it behaves well for symmetric and medium-tailed distributions. Now, following K¨ossler (2006), we may 5

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consider some important modifications of Lepage-type statistic looking into some other optimal choices of T1 and T2 . For simplicity, we only present the score functions and discuss when they are optimal. For a ˇ ak, and Sen (1999) and K¨ossler (2006). detailed derivation of the scores, readers may see H´ajek, Sid´ 2.1.2

The LPGA statistic (short tails)

If the data indicates a symmetric short-tailed distribution, we consider the Lepage-type statistics using Gastwirth score function (LPGA). Gastwirth (1965) first introduced this type of score function. Here we use slight modifications of LPGA statistics for finite sample sizes as in Mukherjee and Sen (2018). When

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N is odd, the LPGA score functions are defined to be

  k − [ N4 ] − 1 if k ≤ [ N4 ] 
1  aLP GA (k) = k − N − [ N4 ] if k ≥ N − [ N4 ] + 1 N +1   0 otherwise

and

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 N   if k ≤ [ N4 ] [ 4 ] + 1 − k
1  bLP GA (k) = k − N − [ N4 ] if k ≥ N − [ N4 ] + 1 N +1   0 otherwise.

When N is even, the LPGA score functions are defined as

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   k − [ N4 ] − 21  
1 aLP GA (k) = k − N − [ N4 ] − N +1   0

1 2

if k ≥ N − [ N4 ] + 1 otherwise

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 1 N   [ 4 ] + 2 − k 
1 bLP GA (k) = k − N − [ N4 ] − N +1   0

1 2

if k ≤ [ N 4+1 ]

if k ≥ N − [ N4 ] + 1 otherwise,

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and

if k ≤ [ N4 ]

where [x] denotes the greatest integer less than or equal to x. With some elementary probability calculus,

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the moments of TLP GA,1 and TLP GA,2 under IC are E(TLP GA,1 |IC) = 0,

V ar(TLP GA,1 |IC) =

 N 3 N 2 N  mn(2[ 4 ] +3[ 4 ] +[ 4 ]) 3N (N −1)(N +1)2 N N  mn[ 4 ](4[ 4 ]2 −1) 6N (N −1)(N +1)2

E(TLP GA,2 |IC) =

 N2 N  n([ 4 ] +[ 4 ]) N (N +1) N  n[ 4 ]2 N (N +1)

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when N is odd when N is even,

when N is odd when N is even

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and V ar(TLP GA,2 |IC) =

 N 3 N 2 N N 4  mn(2(N −3)[ 4 ] +3(N −1)[ 4 ] +N [ 4 ]−3[ 4 ] ) 3N 2 (N −1)(N +1)2 N 3 mn 4N [ ] −N [N ]−6[ N ]4 ) (  4 4 4 2 2 6N (N −1)(N +1)

when N is odd when N is even.

In particular, TLP GA,1 is the optimal statistic for the location problem if the underlying density is uniformlogistic (0.75), see, for example, B¨ uning and K¨ossler (1999) and K¨ossler (2006). For more details, readers may see, Mukherjee and Sen (2018). The LPlog statistic (medium tails)

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2.1.3

Suppose the underlying distribution is logistic. Note that the logistic distribution is a member of symmetric location-scale family with marginally higher tail-weight than normal distribution. The optimal score function for testing the differences in the location parameters of the two independent logistic distributions is given by

2k − 1. N +1

aLP log (k) =

logistic distributions is given by bLP log (k) = −1 −



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Similarly, the optimal score function for testing the differences in the scale parameters of the two independent    2k N +1 − 1 ln −1 . N +1 k

ˇ ak, and Sen (1999). The resulting Lepage-type test is described For more details, readers may see H´ ajek, Sid´

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in K¨ ossler (2006), and it is abbreviated by LPlog. With some elementary probability calculus, we see that the moments of TLP log,1 and TLP log,2 under IC are as follows:

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E(TLP log,1 |IC) = 0, V ar(TLP log,1 |IC) =

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n E(TLP log,2 |IC) = N

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and

N

X mn V ar(TLP log,2 |IC) = N (N − 1) k=1

 1−

N

2 X kln −N − N +1

2k N +1

k=1



ln



N +1−k k

mn , 3(N + 1)





N +1−k k

!

N

X 2 + kln N (N + 1) k=1



N +1−k k

!2

.

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When the data come from close to symmetric medium-tailed population, we often prefer the logistic score function.

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2.1.4

The LPLT statistic (long tails)

If the data indicates a symmetric long-tailed distribution, then we will use the Lepage-type statistic given by K¨ossler (2006). The corresponding score functions are defined by if k < [ N4 ] + 1

4k

N +1    1

−2

bLP LT (k) = 3

if [ N4 ] + 1 ≤ k ≤ [ 34 (N + 1)]

if k > [ 34 (N + 1)], 

2k −1 N +1

2

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aLP LT (k) =

   −1  

− 1.

The score function aLP LT (k) is asymptotically most powerful for the location problem if the underlying density is logistic-double exponential (0.75) and the score function bLP LT (k) is asymptotically most powerful ˇ ak, and Sen for the scale problem if the underlying density is t with two degrees of freedom (cf. H´ajek, Sid´ (1999)). The resulting Lepage-type statistic is abbreviated by LPLT. With some elementary probability

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calculus, the moments of TLP LT,1 and TLP LT,2 under IC are


2mn (7 + 6N − 9N 2 )[ N4 ] + 24N [ N4 ]2 − 16[ N4 ]3 − 2N + 2N 3 , E(TLP LT,1 |IC) = 0, V ar(TLP LT,1 |IC) = 3(N + 1)2 N (N − 1)

2.2

n N



 2 4mn(N 2 − 4) − 2 and V ar(TLP LT,2 |IC) = . N +1 5(N + 1)3

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E(TLP LT,2 |IC) =

Adaptive Lepage-type schemes

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The Lepage-type NSPM schemes for joint monitoring of location and scale do not take into account the information related to the shape of the underlying process density. Note that different score functions return optimal performance for different types of probability densities with different tail-weights. We propose

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to estimate the tail-weight of the underlying distribution from the Phase-I sample. Further, we use this additional information to choose an appropriate score function and implement the corresponding monitoring scheme. Therefore, the proposed scheme is of an adaptive nature. It is designed to minimize the loss of

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information related to the shape of the underlying density. Hogg (1974) developed the idea of adaptive testing of hypothesis. B¨ uning and Thadewald (2000) and K¨ ossler (2006) proposed adaptive two-sample joint location and scale tests. In fact, the ideas of them are

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similar. They primarily considered the case of symmetric distributions. Nevertheless, their studies showed that tests designed for symmetric distributions are also good for slightly asymmetric distributions. In this paper, we follow the idea of K¨ ossler (2006). At first, we classify the type of underlying distribution with respect to a measure of tail-weight. Thereafter, we apply an appropriate Lepage-type statistic for process monitoring. As a measure of tail-weight we choose W =

F −1 (0.95) − F −1 (0.05) . F −1 (0.85) − F −1 (0.15)

ˆ The measure W is location and scale invariant. Replacing the quantile function F −1 (·) by an estimate Q(·),

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ˆ of W can be defined by an estimate W    X − (1 − )(X(2) − X(1) )   (1) ˆ ˆ Q(0.95) − Q(0.05) ˆ = ˆ W with Q(u) = (1 − )X(j) + X(j+1) ˆ ˆ  Q(0.85) − Q(0.15)   X + (X −X ) (m)

where  = m · u +

1 2

(m)

(m−1)

if u < if

1 2m

1 2m

≤u≤

if u >

2m−1 2m

(2)

2m−1 2m ,

− j, j = [m · u + 21 ], and X(i) is the i-th order statistic of the reference sample of size

2.2.1

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m. Next, we introduce the adaptive test considered by K¨ossler (2006).

An Adaptive Lepage-type statistic [LPA as in K¨ ossler (2006)]

K¨ossler (2006) discussed an adaptive Lepage-type test (LPA), which is based on the three aforementioned ˆ be the selector statistic. individual Lepage-type tests, namely, the LPGA, LPlog and LPLT tests. Let W Then the test LPA is defined by ˆ ≤ 1.55 W

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   LPGA if   LPA = LPlog if    LPLT if

ˆ ≤ 1.8 1.55 < W

ˆ > 1.8. W

That means, the LPGA test is selected for short tails, the LPlog test for medium tails, and the LPLT test for long tails. Note that, the LPGA test, the LPlog test, and the LPLT test, respectively, are asymptotically

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most powerful rank tests of location and scale for detecting shifts in short, medium and long-tailed distribution. Actually, the LPA test as in K¨ ossler (2006) is based on the asymptotic theory. To find out whether the asymptotic theory can be applied for moderate or small sample sizes, we investigate the performance of

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the LPA statistic for various distributions in Section 3. The results are very interesting. For small samples, especially when, the size of the test sample is really too small compared to the size of the reference sample, the asymptotic results are not always very accurate. Thus, we propose a new adaptive Lepage-type test

An adaptive Lepage-type scheme with finite sample correction [MLPA]

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2.2.2

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with finite sample correction in the following subsection based on the simulation results.

In most practical situations, the test sample size n is small. On the contrary, the score functions lead to an asymptotically most powerful test for a shift in the location or scale parameters of a given density. The

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exact results for a finite sample, especially when test sample size is small, may slightly vary. Mukherjee and Rakitzis (2018) noted a similar feature even in the context of parametric. They mentioned that even though both the Wald statistic and the likelihood ratio statistic for monitoring multiple parameters are asymptotically equivalent, their finite sample properties are slightly different. We study the exact out-ofcontrol (OOC) performance of the LPGA, LPlog and LPLT statistics and the adaptive-type LPA statistic. Computational details are deferred to the next section. As expected, we observe that the LPlog scheme is the best for the logistic distribution (W = 1.70). Interestingly, we find that the LPlog scheme performs slightly better than the LPGA scheme even when the underlying distribution is Uniform (W = 1.29). Also, the LPlog scheme is slightly better than the LPLT scheme for the Laplace distribution (W = 1.91) for an upward shift in scale. For the downward shift in scale, however, the LPLT scheme is better when the

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underlying process density is Laplace. The LPGA scheme is not so competitive. The LPLT scheme is still very competitive when the underlying process distribution is t with 2 degrees of freedom, hereafter t(2), (W = 2.11) or Cauchy (W = 3.22). Therefore, we update the LPA scheme based on the small sample behavior and design a simplified adaptive scheme based on the LPlog and LPLT schemes only as:  LPlog MLPA = LPLT

ˆ ≤c if W

ˆ > c, if W

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where c is a suitable constant between 1.70 to 2.11 for monitoring a class of symmetric distributions. We refer to the new scheme as Modified LPA (MLPA). Since the value of W of the Laplace distribution falls almost at the center of the interval (1.70, 2.11), we choose it to determine the value of c for a class of symmetric distributions. To this end, we consider a well-known metric, known as the expected weighted run length, to evaluate the overall performance. In the present case, we are monitoring two parameters simultaneously. Therefore, we use a simple uniform weighting scheme in the line of Ryu, Wan, and Kim (2010), Mukherjee and Marozzi (2017b) and Mukherjee and Sen (2018). This simplified index for measuring

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the overall performance of a monitoring scheme is known as Expected Average Run Length (EARL). For symmetric process distributions, one may select the boundary c that minimizes EARL under the Laplace distribution. That is, we propose to minimize, EARL(c|Laplace) =

Z

∞

−∞

Z

∞

ω(θ, δ)ARL(θ, δ|c)dF1 (θ)dF2 (δ),

(3)

0

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where ω(θ, δ) is a suitable weight and F1 and F2 are the cdf of θ and δ respectively. The ARL(θ, δ|c) is the average run length value of the MLPA scheme, under a shift (θ, δ) from a standard Laplace distribution, given the threshold c. We note that for the MLPA scheme the selection of c matters most when 0 ≤ θ ≤ 3

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and 1 ≤ δ ≤ 2. A larger shift in either parameter is almost immediately detected by the MLPA scheme

with any value of c. Therefore, we consider F1 and F2 as uniform over the support of the distribution and ω(θ, δ) = 1. For ease of computation, we may approximate the integrals as in (3) by the Riemann sum.

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We choose c ∈ (1.70, 2.11) that minimizes EARL(c|Laplace). Consequently, a simple EARL minimization, similar to numerical EM algorithm, establishes that c = 1.91.

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Note that, a similar approach may be used to determine the choice threshold c for the right-skewed processes. The OOC-ARL behavior of the log-normal (W = 2.02) distribution is the same as that of the normal distribution, because the ranking is not affected by a monotone transform, for example, the log

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transformation. Similarly, we see that the OOC performance of the pairs, namely, logistic and log-logistic (W = 3.45) distributions, Laplace and log-Laplace (W = 3.26) distributions, t(2) and log-t(2) (W = 4.93) distributions etc are same. While in the first three cases, the LPlog scheme is better; the LPLT scheme is better in the last case. Thus, for right skewed processes, c is 4.19.

2.3

Design and implementation

In this subsection, we discuss the Shewhart-type monitoring scheme based on the LPGA, the LPlog, the LPLT, the LPA and the MLPA statistics. Note that, the Lepage-type statistic T given in (1) is nonnegative by definition. Also, irrespective of the nature of the shift, the expected value of T under OOC, is larger on an average. In other words, T is expected to be larger if there is a shift in location parameter or in 10

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scale parameter or in both. Thus, each of the five monitoring schemes requires only an upper control limit (U CL). 2.3.1

Determination of upper control limits

Consider the Phase-II monitoring scheme based on the Lepage-type statistic TLP [S] , where [S] stands for the corresponding monitoring statistic, for example, [S] = W AB, GA, log, LT etc. Let the U CL corresponding to the scheme TLP [S] be HLP [S] . Let us denote the run-length variable corresponding to the scheme based on TLP [S] by RLLP [S] . The general form of the run-length distribution and the average run length may be

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given by the following Theorem.

Theorem 1 For r = 1, 2, . . . , the distribution of RLLP [S] may be given by

  where Ψ(Xm ) = P rob T1,LP [S] ≤ HLP [S] |Xm . Proof is deferred to Appendix.

As a consequence of Theorem 1, 

E RLLP [S]



∞ X

=

r=1

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  P rob RLLP [S] = r = E [Ψ(Xm )]r−1 − E [Ψ(Xm )]r ,

∞   X r · P rob RLLP [S] = r = E [Ψ (Xm )]r = E r=0



 1 . 1 − Ψ(Xm )

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Note that Ψ (Xm ) is distribution-free when the process is IC but it depends on the underlying cdf under the OOC situation. In general, Theorem 1 is valid under both the IC and OOC situations. Further, as m m+n

→ λ(> 0), a constant, the run-length distribution converges to a geometric

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min(m, n) → ∞, and

distribution. It is easy to see that E[RLLP [S] ] increases as HLP [S] increases, and we fix a target value of   IC ARL (ARL0 )= E RLLP [S] |IC and determine HLP [S] . Moreover, given Xm , W is a constant. As a

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consequence, run length distribution and the expected run-length of the LPA or the MLPA schemes, will follow from their individual schemes.

Implementation steps of the LPGA, LPlog and LPLT schemes

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2.3.2

The Shewhart-type LPGA monitoring scheme, which is a special case of the percentile modified Lepage schemes as in Mukherjee as Sen (2018) and two new monitoring schemes namely, LPlog and LPLT may be

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implemented in practice via the following steps: Step-1: Collect a random sample Xm = (X1 , X2 , . . . , Xm ) of size m and establish it as a reference sample from an IC process through a suitable Phase-I analysis. There are several approaches related to establishment of Phase-I sample, which is out of scope and purview of this article and hence, further details are omitted here. Step-2: Let Yj = (Yj1 , Yj2 , . . . , Yjn ) be the j-th test sample of size n, j = 1, 2, . . . Step-3: (i) For LPGA scheme, compute TLP GA,1j and TLP GA,2j using the reference sample and the j-th test sample, and obtain their means and variances according to whether N = m + n is even or odd, as described in Section 2.1.2. 11

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(ii) For LPlog scheme, calculate TLP log,1j and TLP log,2j using the reference sample and the j-th test sample, and obtain their means and variances, as described in Section 2.1.3. (iii) For LPLT scheme, calculate TLP LT,1j and TLP LT,2j using the reference sample and the j-th test sample, and obtain their means and variances, as described in Section 2.1.4. Step-4: Compute the plotting statistic TLP GA,j or TLP log,j or TLP LT,j for the j-th test sample, j = 1, 2, . . . Step-5: Let HLP GA , HLP log , and HLP LT be the U CL of the LPGA, LPlog and LPLT schemes respectively. [S] = GA or log or LT , as the case may be.

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For j = 1, 2, ..., compare the plotting statistic TLP [S],j with the corresponding U CL, HLP [S] , where

Step-6: We obtain an OOC signal at the j-th stage of inspection if TLP [S],j exceeds HLP [S] , [S] = GA or log or LT , and a search for assignable cause begins. Otherwise, the process is thought to be IC, an monitoring continues to the next test sample. 2.3.3

Implementation steps of the LPA scheme

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The LPA monitoring scheme may be constructed as follows: Step-1 and 2: Same as in Section 2.3.2.

ˆ of tail-weight based on the reference sample Xm = (X1 , X2 , . . . , Xm ) Step-3: Calculate the estimate W

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as in (2), and then select either of the LPGA, LPlog and LPLT statistics for the classified type ˆ ≤ 1.55, we compute TLP GA,j using the reference sample and of distribution. To be precise, if W ˆ > 1.55 and W ˆ ≤ 1.8, we the j-th test sample, and plot TLP GA,j against the U CL HLP GA ; If W calculate TLP log,j using the reference sample and the j-th test sample, and plot TLP log,j against the ˆ > 1.8, we calculate TLP LT,j using the reference sample and the j-th test sample, U CL HLP log ; If W

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and plot TLP LT,j against the U CL HLP LT . Step-4: We obtain an OOC signal at the j-th stage of inspection if TLP [S],j exceeds HLP [S] in which [S]

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stands for either of the GA, log and LT . We recommend stopping production and searching for assignable causes. Otherwise, the process is thought to be IC, and monitoring proceeds to the next

2.3.4

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test sample.

Implementation steps of the MLPA scheme

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The MLPA monitoring scheme may be constructed as follows: Step-1 and 2: Same as in Section 2.3.2. ˆ of tail-weight based on the reference sample Xm = (X1 , X2 , . . . , Xm ) as Step-3: Calculate the estimate W in (2), and then select the LPlog or LPLT statistic for the classified type of distribution. To be ˆ ≤ 1.91, we compute TLP log,j using the reference sample and the j-th test sample, and precise, if W ˆ > 1.91, we calculate TLP LT,j using the reference sample plot TLP log,j against the U CL HLP log ; If W and the j-th test sample, and plot TLP LT,j against the U CL HLP LT . Step-4: We obtain an OOC signal at the j-th stage of inspection if TLP [S],j exceeds HLP [S] in which [S] stands for either of the log and LT . We recommend stopping production and searching for assignable causes. Otherwise, the process is thought to be IC, and monitoring proceeds to the next test sample. 12

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3

Numerical results and comparisons

In this section, we analyse IC and OOC performance of the proposed class of monitoring schemes, including the LPGA, LPlog, LPLT, LPA and the MLPA. The ARL and the standard deviation of the run length (SDRL) are popular performance indicators, but since the run length distribution is right-skewed, it is worthwhile to study various summary measures viz. a number of percentiles including the 5th, 25th, 50th, 75th and the 95th.

IC performance

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3.1

For the implementation of the aforementioned schemes, we need to determine the U CL H that provides certain target ARL0 . We perform a simulation study based on Monte-Carlo in FORTRAN to determine the H values on the basis of 50,000 replications. Some algorithms for computing H of monitoring schemes, see, for example, Capizzi and Masarotto (2016). Because of the distribution-free nature of the proposed class of monitoring schemes, without loss of generality, we generate m observations from a standard normal

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distribution for the Phase-I sample and n observations from the same distribution for each test sample. Note that one can also directly generate ranks from uniform distribution. In Table 1, we provide the H values for some selected (m, n) and nominal ARL0 . We choose m = 50, 100, 150, and 300, to cover small to moderate reference sample sizes. Further, we select the test sample size n = 5 and 11. Finally, we consider the nominal ARL0 as 250, 370 and 500. In NSPM literature, the ARL0 values of 250, 370 and 500 are the three common choices in recent times, see for example, Chowdhury, Mukherjee, and Chakraborti (2014) and Mukherjee and Sen (2018). Original Shewhart X chart based on 3-sigma limits on either side of the

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central line, has an ARL0 = 370.4. That is why, the value 370 is very popular in almost all SPM literature. Nevertheless, the modern industry standard is to set a nominal ARL0 of 500 to minimize early false alarms.

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We provide H for these standard values of ARL0 for implementation of the proposed class of schemes in practice. Following the modern convention, we use ARL0 of 500 for the illustration purpose and also for the follow-up study. We notice that, a large number of existing papers on joint monitoring, for example,

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Mukherjee and Chakraborti (2012) and Chowdhury, Mukherjee, and Chakraborti (2014), considered n = 11 along with n = 5. The original Lepage statistic and the LPGA statistic have different expressions depending on whether m + n is odd or even. This may be a possible reason. One can use the same simplified code

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designed for odd m + n case if m = 100, n = 5 or m = 100, n = 11. We follow the same convention. For a given combination of (m, n), we obtain the H values from Table 1, for a nominal ARL0 of 500, and investigate various characteristics of the IC run length distribution via simulation. In particular, we

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study the attained ARL0 , SDRL0 and standard error of ARL estimation (SE) and mention them in the 1st row of each cell of Table 2. In Table 2, we also present some percentiles (5th, 25th, 50th, 75th, 95th) of the run length distribution in the ascending order in the 2nd row of each cell. To verify the nonparametric nature of the proposed class of monitoring schemes, one can easily check that for a given combination of (m, n; ARL0 ), the same H values as in Table 1 are valid for any other non-normal distributions. We do not create a separate table to display this, as one can see this later from Tables 4-6. We can see that for θ = 0 and δ = 1, the ARLs and the SDRLs and the run length percentiles are almost the same, because of the distribution-free nature of the plotting statistic under IC set-up. From Table 2, for each scheme, we see that the nominal ARL0 of 500 is much higher than the medians for all combinations of (m, n). The 95th percentile is about 3.2-5.2 times the nominal ARL0 of 500. These 13

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Table 1: The upper control limits H of the LPGA, LPlog and LPLT schemes for some standard values of ARL0 .

LPlog

250 10.17 8.11 11.97 9.81 12.81 10.69 13.44 11.72 9.61 8.29 11.21 9.61 11.97 10.32 13.06 11.26 8.72 9.17 9.63 9.64 9.93 9.94 10.25 10.21

ARL0 370 500 11.02 11.879 8.39 9.03 12.91 13.821 10.65 11.44 13.81 14.75 11.62 12.43 14.56 15.50 12.78 13.71 10.39 11.03 8.74 9.19 12.22 12.96 10.32 10.94 12.98 13.91 11.21 11.89 14.18 15.12 12.29 13.15 9.36 9.91 9.66 10.06 10.32 10.85 10.24 10.69 10.63 11.22 10.51 11.02 11.01 11.61 10.99 11.50

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LPLT

n 5 11 5 11 5 11 5 11 5 11 5 11 5 11 5 11 5 11 5 11 5 11 5 11

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LPGA

m 50 50 100 100 150 150 300 300 50 50 100 100 150 150 300 300 50 50 100 100 150 150 300 300

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Proposed schemes

n H 5

11.879

50

11

9.03

100

5

13.821

11

11.44

100 150

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50

5

14.75

11

12.43

300

5

15.50

300

11

13.71

150

LPGA ARL0 (SDRL0 , SE) percentiles 529.22( 925.65,4.14) 9, 60, 188, 529, 2392 495.82( 989.20,4.42) 6, 35, 124, 418, 2617 507.01( 729.25,3.26) 15, 93, 250, 611, 1843 500.21( 845.58,3.78) 11, 65, 192, 526, 2096 505.03( 675.91,3.02) 18, 106, 276, 625, 1756 495.51( 755.71,3.38) 14, 82, 227, 570, 1908 499.86( 575.69,2.57) 22, 127, 311, 657, 1627 504.97( 663.20,2.97) 19, 108, 277, 635, 1767

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m

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Table 2: IC run-length characteristics of various Lepage-type monitoring schemes for target ARL0 =500.

H

11.03 9.19 12.96 10.94 13.91 11.89 15.12 13.15

LPlog ARL0 (SDRL0 ,SE) percentiles 495.06( 885.22,3.96) 9, 56, 175, 494, 2163 500.51( 942.82,4.22) 6, 43, 144, 465, 2435 497.45( 775.22,3.47) 13, 79, 225, 560, 1925 501.15( 845.99,3.78) 10, 63, 190, 531, 2137 497.95( 713.10,3.19) 15, 92, 247, 594, 1827 504.26( 793.12,3.55) 13, 77, 221, 572, 1982 495.30( 630.67,2.82) 19, 109, 282, 634, 1689 500.83( 693.07,3.10) 17, 100, 260, 607, 1800

H 9.91 10.06 10.85 10.69 11.22 11.02 11.61 11.50

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LPLT ARL0 (SDRL0 ,SE) percentiles 501.18(828.76,3.71) 12, 69, 203, 543, 2096 498.18( 713.55,3.19) 11, 76, 227, 612, 1949 497.91( 715.33,3.20) 15, 92, 249, 595, 1830 501.36( 704.06,3.15) 14, 88, 245, 620, 1867 506.20( 660.78,2.96) 20, 109, 280, 636, 1765 496.74( 672.31,3.01) 16, 97, 261, 623, 1795 495.95( 574.78,2.57) 22, 123, 306, 659, 1595 499.26( 619.68,2.77) 20, 116, 296, 637, 1679

LPA ARL0 (SDRL0 ,SE) percentiles 493.10( 871.66,3.90) 9, 58, 175, 494, 2178 487.47( 927.86,4.15) 6, 42, 139, 461, 2316 515.50( 777.89,3.48) 14, 86, 241, 600, 1948 503.97( 844.67, 3.78) 11, 66, 196, 537, 2128 513.95( 722.34,3.23) 17, 101, 262, 617, 1871 511.23( 794.09,3.55) 14, 82, 228, 572, 2017 512.20( 633.04,2.83) 22, 118, 304, 654, 1702 512.58( 687.38,3.07) 18, 104, 277, 637, 1819

MLPA ARL0 (SDRL0 ,SE) percentiles 479.29( 869.56,3.89) 8, 54, 164, 467, 2109 493.15( 928.30, 4.15) 6, 43, 144, 465, 2344 497.75( 787.13,3.52) 13, 79, 220, 550, 1967 503.70( 854.27,3.82) 11, 65, 194, 528, 2125 492.16( 700.55,3.13) 15, 91, 247, 593, 1789 492.06( 772.80,3.46) 12, 79, 216, 550, 1973 493.57( 631.99,2.83) 19, 110, 282, 624, 1687 496.86( 686.20,3.07) 18, 101, 261, 607, 1786

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results show that the IC run length distribution of the proposed class of schemes is heavily skewed. For a fixed n, all the lower order run-length percentiles increase with an increase in the reference sample size m, however, the higher order run length percentiles, for example, the 95th percentile and the SDRL decrease. On the other hand, for a given m, the opposite is true. That is, as n increases, all the lower order percentiles decrease, but the 95th percentile and the SDRL increase, except for the two cases connected to the LPLT scheme, first when m = 50 and when m = 100. These findings suggest that larger m will greatly reduce early false alarms and eliminate very long runs. Further, the results suggest that the larger test sample size

3.2

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should be avoided.

Comparisons of early false alarms

Too many early false alarms in monitoring a production process are certainly not good from the cost perspective. The IC run length distribution is considered to be satisfactory if it is close to the geometric distribution (Hawkins and Olwell (1998)). Note that when the IC run length distribution is the geometric distribution, the false alarm rate on or before 25th test samples is about 0.0488. Therefore, we also compare

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the false alarm rates for the first 25 test samples for the proposed class of schemes. For a fair comparison, all the monitoring schemes are calibrated so that they have ARL0 values of approximately 500. The results are shown in Table 3. We see that for all schemes considered, as m increases from 50 to 300, for a fixed n, these probabilities of false alarms for the first 25 test samples gradually decrease, but for a fixed m, these probabilities increase as n increases, except for some sampling fluctuations. For m = 300 and n = 5, the false alarm rate for the proposed class of monitoring scheme is close to 5%. We may conclude that for small

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n, when m ≥ 100, the early false alarm rates are reasonable for all the schemes considered in this paper.

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Table 3: Proportion of false alarms for various Lepage-type monitoring schemes before 25 test samples when target ARL0 =500. LPGA 0.118 0.080 0.068 0.058 0.196 0.113 0.091 0.067

LPlog 0.131 0.091 0.079 0.064 0.171 0.121 0.094 0.076

LPLT 0.111 0.077 0.069 0.056 0.103 0.091 0.076 0.063

LPA 0.128 0.084 0.073 0.064 0.173 0.114 0.090 0.067

MLPA 0.134 0.095 0.078 0.065 0.175 0.121 0.096 0.071

OOC performance comparisons

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3.3

n 5 5 5 5 11 11 11 11

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m 50 100 150 300 50 100 150 300

The IC performance of the proposed class of monitoring schemes is robust for all univariate and continuous process distributions. However, this is not true for the OOC performance. Thus, it is important to examine their OOC performance for a particular distribution to assess the schemes’ efficacy in a given realistic situation. A monitoring scheme is considered to have better performance than its competitors if it has the smaller OOC ARL (ARL1 ) value when ARL0 is the same for all the competitive schemes. We conduct a simulation study based on Monte-Carlo to evaluate the performance of the proposed class of schemes. As a competitive scheme, we also include the SC scheme, based on the Cucconi statistic and proposed by Chowdhury, Mukherjee, and Chakraborti (2014), as it is well-known to be superior in many contexts. The Cucconi statistic addresses the location-scale problem by considering the squares of ranks and ‘contrary 15

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ranks’. Marozzi (2009) performed a detailed power simulation study including distributions of different shapes and showed that the Cucconi test performs as good or better than the classical Lepage test in many situations. Interested readers may also see Marozzi (2013). Most of the recent literature on the joint monitoring of location and scale acknowledged the robust performance of the SC chart as compared to the SL chart. Note that, we do not include any linear rank test, designed for asymmetric distributions, in our adaptive schemes. Therefore, at the very outset, we consider four symmetric distributions from the location-scale family. They are as follows:

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i. Thin-tailed Uniform distribution over support (θ − δ, θ + δ). The process mean and variance are respectively, θ and δ 2 /3. We denote this by Uniform(θ, δ).

ii. Medium-tailed normal distribution with mean θ and standard deviation δ, denoted by N (θ, δ). √ iii. Moderately heavy-tailed Laplace distribution with mean θ and standard deviation δ 2, denoted by Laplace(θ, δ). Finally,

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iv. Very heavy-tailed Cauchy distribution, denoted by Cauchy(θ, δ), for which mean and variance do not exist.

When the process is IC, we set θ = 0 and δ = 1 in all four situations. That is, under IC, we consider samples from the standard form of the distributions. We do not restrict our simulation study only within the above four symmetric distributions. We also attempt to answer the question of how the proposed class of schemes and their competitors behave in the case of the skewed distributions. To this end, we choose two

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asymmetric distributions. The Gumbel distribution (denoted by Gumbel(θ, δ)) and two-parameter shifted exponential distribution (denoted by SE(θ, δ)) represent the skewed medium-tailed distributions. For the Gumbel(θ, δ) distribution, the IC samples are taken from Gumbel(0,1), while the test samples are coming −z

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from a Gumbel(θ, δ) distribution with pdf f (x) = 1δ e−z−e , z =

x−θ δ .

For the SE(θ, δ) distribution, the IC

samples are drawn from SE(0,1), whereas the test samples are selected from a SE(θ, δ) distribution having 1

pdf f (x) = 1δ e− δ (x−θ) , x ∈ (θ, ∞) with mean=θ + δ and variance=δ 2 .

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We tabulate the OOC run length properties for different distributions and under various shifts in the

location and scale parameters in Tables 4-6. For illustration, we consider 16 combinations of θ and δ, that

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is, θ = 0, 0.5, 1 and 2 along with δ = 1, 1.25, 1.5 and 2, for m = 100 and n = 5. Note that, we fix the ARL0 of all the competitive schemes to a nominal value of approximately 500. At present, the industry standard is to choose ARL0 = 500. The first row of each cell in Tables 4 to 6 shows the ARL followed by

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the corresponding SDRL in parentheses, whereas the second row shows the values of the 5th, 25th, 50th, 75th and 95th percentiles (in this order). Two types of shades are used in these tables. We highlight the schemes with the best OOC performance in the sense that the least ARL1 is observed for a given shift with a dark gray shade and use a light gray shade for its nearest rival. From Tables 4-6, we may reach at the following conclusions: i. When the process follows the Uniform distribution, the overall performance of the LPlog and MLPA schemes are the best. In fact, their performance is very similar. Apart from that, the LPGA and LPA schemes perform better than the other competing schemes for all shifts considered. This is somewhat surprising. Ideally, the LPGA scheme should offer the best performance here, since the Uniform distribution has a very low tail-weight. This is likely to be a finite sample phenomenon, which has also 16

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occurred in B¨ uning and Thadewald (2000). For this reason, we actually propose the new MLPA scheme only with the LPlog and LPLT statistics. Thus, the MLPA scheme may be regarded as an adaptive Lepage-type scheme with finite sample correction. ii. When the underlying process distribution is normal, the LPlog and MLPA schemes are far superior than the competing schemes for detecting a shift in the scale parameter. This is expected as the normal distribution is symmetric and has a medium tail-weight, for which the LPlog test offers relatively higher power. When there is a small to moderate location shift but there is no scale shift, the SL scheme

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performs the best. iii. When the process follows the Laplace distribution, the LPlog scheme performs the best and the MLPA scheme emerges as the closest competitor when the shift is predominantly in the scale parameter. When only the location shift takes place and the scale parameter remains invariant, the SL scheme performs the best. The SL scheme also gives better result for moderate to large shifts in location, with small shifts in scale. However, when the magnitude of scale shift increases, we observe that the LPlog and

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MLPA schemes detect small shifts in the location parameter faster than their competitors whereas for a larger amount of location shifts, all the schemes perform almost similarly. The tail-weight of the Laplace distribution is 1.91 which is marginally higher than 1.8. Ideally, the LPLT scheme should have offered much better performance in this situation. However, we see a difference. This is again likely to be a finite sample phenomenon.

iv. For the Cauchy distribution, the tail-weight is 3.22 which is much larger than 1.8. As expected, the

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LPLT scheme for the heavy tailed distributions offers the best performance when the shift is mainly in the scale parameter. Consequently, both the LPA and MLPA schemes have a very similar performance as that of the LPLT scheme. The SL scheme is the best in detecting a shift in the location parameter when

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there is no shift in the scale parameter. Similarly, for moderate to large location shifts accompanied by some small scale shifts, the SL scheme performs better than its competitors. For small location shifts along with moderate to large scale shifts, however, the LPLT, LPA and MLPA schemes perform the

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best.

v. For the Gumbel distribution and the shifted exponential distribution, the LPlog and MLPA schemes

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perform more or less better than the other schemes when the shift is mainly in the scale parameter. When the shift is mainly in the location parameter, the SL scheme performs the best. Note that, for skewed distributions, the superiority of the proposed schemes is not obvious. It is not surprising, since

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our focus here is mainly on the case of symmetric distributions.

In summary, the results from Tables 4-6 are very clear. Among all the schemes considered, we see that the

proposed MLPA scheme is always either the best, or the nearest rival of the best available scheme, especially when the process distribution is symmetric and the shift is predominantly in the scale parameter. Moreover, we also consider larger m and n, such as m = 500 and 1000 and n = 11 and 25. The general patterns in the ARL1 values for larger m or n are quite similar to the case of m = 100 and n = 5 for all schemes. Nevertheless, the OOC performance of these schemes improves as the size of the reference sample or the test sample increases. For example, we represent ARL curves of the MLPA scheme for (m, n) = (100, 5) and (m, n) = (100, 11) in Fig. 1 for comparison purposes. In Fig. 1, we present a pure location shift case (left

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panel), a pure scale shift case (middle panel) and a mixed location-scale shift case (δ = eθ , right panel). We see from Fig. 1 that for m = 100, the ARL1 decreases as n increases from 5 to 11, for all distributions and all types shifts under investigation. The design of the above schemes focuses on optimizing its performance under simultaneous locationscale shifts. However, it may happen that the location and scale parameters remain in IC, but the shape 
ϑ parameter keeps on changing. A classic case is when the shifted process model is G(x) = F x−θ , where δ the constant ϑ represents the shape parameter. Under IC situation, we expect: θ = 0, δ = 1 and ϑ = 1. To examine the effect of shifts in location, scale and shape parameters, we consider two choices of θ as 0 and

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1 and two choices of δ as 1 and 2. Further, we consider three choices of ϑ as 0.5, 1 and 5. Altogether, we consider 12 triplets of θ, δ and ϑ values, for performance analysis. For brevity, we only tabulate the results for m = 100, n = 5, ARL0 = 500. Here we restrict ourselves only to three process distributions, namely, skew-normal; skew Cauchy and three-parameter exponential. The results are shown in Table 7. We see from Table 7 that the joint monitoring schemes can also detect a shift in the shape parameter. When ϑ = 1, G(x) boils down to a classical location-scale model. Consequently, the results corresponding to ϑ = 1, are

AN US

same as the analogous cases in Tables 4-6.

We see from Table 7, that for the skew-normal distribution, the LPlog and MLPA schemes perform the best when ϑ = 0.5. When ϑ = 5 and there is a location or scale shift, all schemes perform similarly. Nevertheless, if location and scale remain invariant and ϑ undergoes an upward shift to 5, the SL scheme performs the best. For the skew-Cauchy distribution, we see that the LPlog scheme performs the best and the LPGA scheme is the closest competitor when ϑ = 0.5. When ϑ = 5, the SL scheme performs the best. For the three-parameter exponential distribution, when ϑ = 0.5, the LPlog and MLPA schemes outperform

M

all other schemes if the location parameter remains invariant. Otherwise, the SL scheme is the best. The SL scheme is also the best when ϑ shifts to 5 but location and scale parameters remain invariant. Finally, if

ED

the upward shift in ϑ happens along with a location or scale change, we see that all the schemes are almost

1

1.5 θ

2

3

1

1.49

1.82 δ

2.23

ARL 100 200 300 400 500 2.72

1.5 θ

2

(100,11)

3

1

1.49

1.82 δ

ARL 100 200 300 400 500

2.23

2.72

0.5

1 1.5 2 θ location alternative

3

1

1.25

1.5

(100,11)

0

0.25

0.5

0.75 θ

1

1.25

1.5

SE(θ, δ)

(100,5) (100,11)

0

0

200

ARL 400

(100,11)

0.75 θ

(100,5) (100,11)

0

(100,5)

0.5

(100,5)

SE(θ, δ)

800

SE(θ, δ)

0

0.25

ARL 100 200 300 400 500

1

0

Cauchy(θ, δ)

(100,5)

0

(100,11)

0.5

(100,11)

0

(100,5)

0 0

(100,5)

Cauchy(θ, δ)

ARL 100 200 300 400 500

ARL 100 200 300 400 500

Cauchy(θ, δ)

600

AC

CE

0.5

(100,11)

0

0

PT

(100,11)

N (θ, δ)

(100,5)

0

(100,5)

0

N (θ, δ)

ARL 100 200 300 400 500

ARL 100 200 300 400 500

N (θ, δ)

ARL 100 200 300 400 500

equally good.

1

1.49

1.82 2.23 δ scale alternative

2.72

0

0.25

0.5 0.75 1 θ mixed alternative

1.25

1.5

Figure 1: ARL curves of the MLPA scheme under different distributions for (m, n) = (100, 5) and (m, n) = (100, 11). 18

ACCEPTED MANUSCRIPT

Table 4: Comparisons of various Shewhart charts under Uniform(θ, δ) and N (θ, δ) distributions for m=100, n=5 and ARL0 =500.

1.5

0

2

0.5

1

0.5 1.25 0.5

1.5

0.5

2

1

1

1

1.25

1

1.5

1

2

2

1

2

1.25

2

1.5

2

2

θ

δ

0

1

0

1.25

0

1.5

0

2

0.5

1

0.5 1.25 0.5

1.5

0.5

2

1

1

1

1.25

1

1.5

1

2

2

1

2

1.25

2

1.5

2

2

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0

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1.25

M

0

ED

1

PT

0

Uniform(θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 515.41( 755.03) 500.19( 786.21) 508.65( 727.44) 510.21( 741.63) 505.43( 674.15) 493.89( 695.45) 500.78( 785.50) 16, 94, 255, 603, 1935 13, 79, 221, 565, 1973 17, 95, 255, 611, 1873 16, 93, 252, 598, 1921 18, 105, 273, 631, 1770 16, 93, 251, 594, 1781 13, 80, 222, 562, 1969 19.48( 19.65) 8.28( 7.91) 21.56( 22.29) 19.31( 19.70) 39.35( 41.50) 22.84( 23.99) 8.21( 7.88) 1, 6, 13, 27, 58 1, 3, 6, 11, 24 2, 6, 15, 29, 65 1, 6, 13, 26, 58 2, 11, 26, 53, 122 2, 7, 15, 31, 70 1, 3, 6, 11, 24 7.21( 6.79) 3.36( 2.82) 7.53( 7.20) 7.21( 6.84) 14.66( 14.80) 8.20( 7.83) 3.39( 2.88) 1, 2, 5, 10, 21 1, 1, 2, 4, 9 1, 2, 5, 10, 22 1, 2, 5, 10, 21 1, 4, 10, 20, 44 1, 3, 6, 11, 24 1, 1, 2, 4, 9 3.18( 2.66) 1.75( 1.16) 3.13( 2.59) 3.20( 2.65) 5.82( 5.40) 3.35( 2.84) 1.74( 1.14) 1, 1, 2, 4, 8 1, 1, 1, 2, 4 1, 1, 2, 4, 8 1, 1, 2, 4, 8 1, 2, 4, 8, 16 1, 1, 2, 4, 9 1, 1, 1, 2, 4 6.30( 5.94) 5.49( 5.08) 11.87( 12.25) 6.26( 5.96) 11.09( 11.36) 9.48( 9.50) 5.56( 5.14) 1, 2, 4, 8, 18 1, 2, 4, 7, 16 1, 4, 8, 16, 36 1, 2, 4, 8, 18 1, 3, 8, 15, 33 1, 3, 6, 13, 28 1, 2, 4, 7, 16 4.61( 4.07) 4.32( 3.82) 9.19( 8.89) 4.67( 4.16) 8.87( 8.69) 7.69( 7.34) 4.30( 3.81) 1, 2, 3, 6, 13 1, 2, 3, 6, 12 1, 3, 6, 12, 27 1, 2, 3, 6, 13 1, 3, 6, 12, 26 1, 3, 5, 10, 22 1, 2, 3, 6, 12 3.81( 3.25) 3.28( 2.78) 6.31( 5.85) 3.79( 3.31) 7.36( 6.97) 5.76( 5.32) 3.27( 2.77) 1, 1, 3, 5, 10 1, 1, 2, 4, 9 1, 2, 4, 8, 18 1, 1, 3, 5, 10 1, 2, 5, 10, 21 1, 2, 4, 8, 16 1, 1, 2, 4, 9 2.65( 2.12) 1.74( 1.14) 2.99( 2.46) 2.63( 2.07) 4.69( 4.23) 3.10( 2.61) 1.74( 1.14) 1, 1, 2, 3, 7 1, 1, 1, 2, 4 1, 1, 2, 4, 8 1, 1, 2, 3, 7 1, 2, 3, 6, 13 1, 1, 2, 4, 8 1, 1, 1, 2, 4 1.64( 1.03) 1.47( 0.85) 1.98( 1.45) 1.64( 1.03) 1.97( 1.43) 1.80( 1.25) 1.47( 0.84) 1, 1, 1, 2, 4 1, 1, 1, 2, 3 1, 1, 1, 2, 5 1, 1, 1, 2, 4 1, 1, 1, 2, 5 1, 1, 1, 2, 4 1, 1, 1, 2, 3 1.72( 1.12) 1.62( 1.01) 2.47( 1.96) 1.73( 1.13) 2.44( 1.91) 2.20( 1.66) 1.62( 1.01) 1, 1, 1, 2, 4 1, 1, 1, 2, 4 1, 1, 2, 3, 6 1, 1, 1, 2, 4 1, 1, 2, 3, 6 1, 1, 2, 3, 6 1, 1, 1, 2, 4 1.80( 1.21) 1.72( 1.12) 2.80( 2.27) 1.79( 1.19) 2.74( 2.22) 2.51( 1.97) 1.72( 1.12) 1, 1, 1, 2, 4 1, 1, 1, 2, 4 1, 1, 2, 4, 7 1, 1, 1, 2, 4 1, 1, 2, 4, 7 1, 1, 2, 3, 6 1, 1, 1, 2, 4 1.82( 1.22) 1.70( 1.10) 2.76( 2.25) 1.82( 1.22) 3.01( 2.51) 2.54( 2.00) 1.71( 1.10) 1, 1, 1, 2, 4 1, 1, 1, 2, 4 1, 1, 2, 4, 7 1, 1, 1, 2, 4 1, 1, 2, 4, 8 1, 1, 2, 3, 7 1, 1, 1, 2, 4 1.00( 0.00) 1.00( 0.00) 1.00( 0.00) 1.00( 0.00) 1.00( 0.00) 1.00( 0.00) 1.00( 0.00) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1.00( 0.01) 1.00( 0.00) 1.00( 0.00) 1.00( 0.00) 1.00( 0.01) 1.00( 0.00) 1.00( 0.00) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1.01( 0.10) 1.00( 0.05) 1.00( 0.06) 1.01( 0.10) 1.00( 0.06) 1.00( 0.07) 1.00( 0.05) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1.08( 0.29) 1.06( 0.26) 1.18( 0.48) 1.08( 0.29) 1.18( 0.47) 1.15( 0.42) 1.06( 0.25) 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 N (θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 507.01( 729.25) 497.45( 775.22) 497.91( 715.33) 515.50( 777.89) 496.73( 656.78) 495.56( 703.21) 497.75( 787.13) 15, 93, 250, 611, 1843 13, 79, 225, 560, 1925 15, 92, 249, 595, 1830 14, 86, 241, 600, 1948 17, 101, 271, 622, 1743 16, 94, 249, 591, 1854 13, 79, 220, 550, 1967 73.74( 97.81) 51.61( 73.51) 72.89( 95.45) 63.38( 87.75) 102.39( 122.23) 74.27( 94.49) 51.16( 71.23) 3, 17, 43, 93, 242 2, 11, 28, 63, 176 3, 17, 43, 92, 241 3, 14, 35, 78, 219 5, 25, 62, 134, 334 3, 18, 44, 94, 246 2, 12, 29, 63, 173 23.34( 26.81) 14.05( 16.38) 23.18( 26.14) 18.80( 22.74) 37.47( 42.32) 24.38( 27.50) 14.19( 16.63) 1, 6, 15, 31, 74 1, 4, 9, 18, 45 2, 6, 15, 30, 73 1, 5, 11, 24, 61 2, 10, 24, 50, 120 2, 7, 15, 32, 77 1, 4, 9, 18, 45 6.92( 6.91) 3.98( 3.75) 6.69( 6.66) 5.40( 5.55) 11.69( 11.86) 7.14( 7.05) 4.07( 3.88) 1, 2, 5, 9, 20 1, 1, 3, 5, 11 1, 2, 5, 9, 20 1, 2, 4, 7, 16 1, 3, 8, 16, 35 1, 2, 5, 9, 21 1, 1, 3, 5, 12 78.13( 140.81) 82.61( 162.08) 79.39( 136.09) 80.31( 154.01) 68.84( 108.32) 70.87( 119.66) 81.36( 150.02) 3, 14, 36, 85, 283 3, 13, 36, 88, 309 3, 15, 39, 92, 280 3, 14, 37, 87, 289 3, 14, 35, 81, 242 3, 14, 34, 81, 254 3, 13, 36, 88, 294 25.66( 33.92) 21.05( 28.30) 28.36( 35.34) 23.74( 31.21) 30.96( 39.16) 26.26( 33.99) 21.35( 28.56) 1, 6, 15, 32, 85 1, 5, 12, 26, 70 2, 7, 17, 36, 93 1, 6, 14, 30, 78 2, 8, 19, 39, 100 2, 7, 16, 33, 86 1, 5, 12, 27, 70 12.67( 14.22) 9.19( 10.32) 14.19( 15.64) 10.99( 12.24) 18.25( 20.56) 13.59( 15.07) 9.25( 10.37) 1, 4, 8, 16, 39 1, 3, 6, 12, 28 1, 4, 9, 19, 44 1, 3, 7, 14, 34 1, 5, 12, 24, 58 1, 4, 9, 18, 42 1, 3, 6, 12, 28 5.48( 5.28) 3.49( 3.11) 5.62( 5.39) 4.51( 4.44) 8.64( 8.46) 5.84( 5.62) 3.56( 3.31) 1, 2, 4, 7, 16 1, 1, 2, 5, 10 1, 2, 4, 7, 16 1, 2, 3, 6, 13 1, 3, 6, 12, 25 1, 2, 4, 8, 17 1, 1, 2, 5, 10 8.50( 10.96) 8.18( 11.39) 8.60( 10.29) 8.39( 11.19) 7.72( 9.40) 7.73( 9.42) 7.88( 10.02) 1, 2, 5, 10, 28 1, 2, 5, 10, 27 1, 2, 5, 11, 27 1, 2, 5, 10, 27 1, 2, 5, 10, 24 1, 2, 5, 10, 24 1, 2, 5, 10, 25 6.02( 6.55) 5.49( 6.03) 7.01( 7.46) 5.89( 6.38) 6.77( 7.17) 6.20( 6.47) 5.50( 5.86) 1, 2, 4, 8, 18 1, 2, 4, 7, 16 1, 2, 5, 9, 21 1, 2, 4, 8, 18 1, 2, 4, 9, 20 1, 2, 4, 8, 19 1, 2, 4, 7, 17 4.83( 4.80) 4.08( 3.94) 5.76( 5.72) 4.59( 4.52) 6.03( 6.03) 5.21( 5.10) 4.13( 4.00) 1, 2, 3, 6, 14 1, 1, 3, 5, 12 1, 2, 4, 8, 17 1, 2, 3, 6, 13 1, 2, 4, 8, 17 1, 2, 4, 7, 15 1, 1, 3, 5, 12 3.42( 3.02) 2.57( 2.11) 3.87( 3.49) 3.02( 2.70) 4.95( 4.61) 3.78( 3.44) 2.61( 2.14) 1, 1, 2, 4, 9 1, 1, 2, 3, 7 1, 1, 3, 5, 11 1, 1, 2, 4, 8 1, 2, 3, 7, 14 1, 1, 3, 5, 11 1, 1, 2, 3, 7 1.22( 0.55) 1.18( 0.50) 1.25( 0.58) 1.21( 0.53) 1.23( 0.55) 1.21( 0.52) 1.18( 0.49) 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1.36( 0.73) 1.29( 0.63) 1.46( 0.85) 1.33( 0.71) 1.42( 0.79) 1.37( 0.74) 1.29( 0.64) 1, 1, 1, 2, 3 1, 1, 1, 1, 3 1, 1, 1, 2, 3 1, 1, 1, 1, 3 1, 1, 1, 2, 3 1, 1, 1, 2, 3 1, 1, 1, 1, 3 1.46( 0.84) 1.37( 0.73) 1.62( 1.03) 1.43( 0.82) 1.60( 1.00) 1.51( 0.90) 1.37( 0.73) 1, 1, 1, 2, 3 1, 1, 1, 2, 3 1, 1, 1, 2, 4 1, 1, 1, 2, 3 1, 1, 1, 2, 4 1, 1, 1, 2, 3 1, 1, 1, 2, 3 1.58( 0.97) 1.43( 0.80) 1.80( 1.24) 1.52( 0.93) 1.90( 1.34) 1.71( 1.13) 1.44( 0.82) 1, 1, 1, 2, 4 1, 1, 1, 2, 3 1, 1, 1, 2, 4 1, 1, 1, 2, 3 1, 1, 1, 2, 5 1, 1, 1, 2, 4 1, 1, 1, 2, 3

CE

δ

AC

θ

19

ACCEPTED MANUSCRIPT

Table 5: Comparisons of various Shewhart charts under Laplace(θ, δ) and Cauchy(θ, δ) distributions for m=100, n=5 and ARL0 =500.

CE

PT

ED

AN US

M

LPGA LPlog LPLT 509.27( 745.47) 502.86( 785.92) 503.06( 717.61) 15, 93, 250, 599, 1950 13, 78, 222, 565, 1992 16, 95, 255, 604, 1859 120.93( 171.39) 97.54( 152.93) 121.00( 165.57) 5, 26, 67, 150, 410 4, 19, 49, 115, 343 5, 27, 67, 151, 411 46.66( 59.41) 32.82( 44.71) 45.81( 56.91) 2, 11, 27, 59, 155 2, 8, 19, 41, 110 2, 11, 27, 59, 149 14.40( 16.29) 9.01( 10.19) 13.84( 15.20) 1, 4, 9, 19, 45 1, 3, 6, 12, 28 1, 4, 9, 18, 43 243.89( 467.43) 222.82( 444.96) 179.16( 349.27) 5, 32, 94, 249, 961 5, 29, 83, 228, 870 5, 27, 75, 189, 660 69.58( 109.91) 57.03( 102.24) 61.65( 94.02) 3, 13, 34, 81, 254 2, 11, 27, 65, 206 3, 12, 32, 74, 216 31.45( 41.44) 23.07( 32.22) 28.91( 37.69) 2, 7, 18, 39, 106 1, 5, 13, 28, 78 2, 7, 17, 36, 97 11.61( 13.28) 7.55( 8.54) 10.88( 12.12) 1, 3, 7, 15, 37 1, 2, 5, 10, 24 1, 3, 7, 14, 34 43.19( 114.45) 39.62( 98.55) 24.14( 47.22) 1, 6, 16, 40, 158 1, 5, 15, 38, 146 1, 4, 11, 26, 87 20.21( 35.44) 17.52( 30.07) 15.23( 23.13) 1, 4, 10, 22, 72 1, 4, 9, 20, 62 1, 3, 8, 18, 51 12.50( 17.84) 10.17( 13.62) 10.50( 13.35) 1, 3, 7, 15, 42 1, 3, 6, 12, 33 1, 3, 6, 13, 34 6.77( 7.54) 4.92( 5.31) 6.22( 6.68) 1, 2, 4, 9, 21 1, 2, 3, 6, 15 1, 2, 4, 8, 19 2.16( 2.63) 2.39( 2.92) 1.90( 1.61) 1, 1, 1, 2, 6 1, 1, 1, 3, 7 1, 1, 1, 2, 5 2.16( 2.25) 2.29( 2.27) 2.09( 1.75) 1, 1, 1, 2, 6 1, 1, 2, 3, 6 1, 1, 1, 3, 5 2.19( 2.08) 2.21( 2.06) 2.18( 1.79) 1, 1, 1, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 2.16( 1.83) 2.00( 1.60) 2.22( 1.80) 1, 1, 2, 3, 6 1, 1, 1, 2, 5 1, 1, 2, 3, 6

AC

δ

CR IP T

Laplace(θ, δ) LPA SL SC MLPA 485.38( 704.18) 497.38( 659.41) 493.71( 701.40) 486.51( 719.65) 0 1 14, 86, 230, 558, 1795 18, 104, 273, 621, 1746 16, 92, 247, 592, 1814 13, 82, 224, 553, 1775 109.54( 153.68) 154.12( 197.71) 121.52( 163.67) 105.27( 151.64) 0 1.25 4, 23, 60, 135, 382 7, 36, 91, 195, 516 5, 27, 68, 152, 416 4, 21, 55, 129, 369 40.82( 52.83) 66.99( 79.55) 48.02( 60.72) 37.96( 50.22) 0 1.5 2, 9, 24, 52, 137 3, 17, 41, 88, 215 2, 12, 28, 61, 159 2, 9, 21, 48, 127 12.18( 14.32) 23.07( 25.13) 14.59( 16.12) 11.26( 12.98) 0 2 1, 3, 8, 16, 39 1, 6, 15, 31, 72 1, 4, 9, 19, 45 1, 3, 7, 15, 36 182.10( 359.14) 159.25( 293.47) 178.89( 333.05) 189.86( 377.48) 0.5 1 5, 26, 73, 186, 693 4, 24, 68, 173, 607 4, 26, 74, 190, 697 5, 26, 74, 193, 733 58.46( 95.42) 65.23( 93.54) 61.10( 96.33) 55.69( 89.37) 0.5 1.25 2, 11, 29, 69, 206 3, 14, 34, 78, 231 2, 12, 31, 72, 212 2, 11, 28, 65, 198 26.60( 35.68) 35.84( 45.72) 29.44( 39.68) 24.84( 33.27) 0.5 1.5 1, 6, 15, 33, 89 2, 8, 21, 46, 119 2, 7, 17, 36, 99 1, 6, 14, 31, 84 9.87( 11.29) 16.38( 18.52) 11.40( 12.62) 9.05( 10.60) 0.5 2 1, 3, 6, 13, 31 1, 5, 10, 22, 51 1, 3, 7, 15, 35 1, 3, 6, 12, 29 28.47( 71.93) 20.05( 38.47) 25.96( 59.39) 30.00( 74.85) 1 1 1, 5, 11, 28, 101 1, 4, 9, 22, 71 1, 4, 11, 27, 94 1, 5, 12, 29, 111 15.21( 23.34) 14.04( 21.07) 15.30( 24.06) 15.41( 25.26) 1 1.25 1, 3, 8, 18, 52 1, 3, 8, 17, 47 1, 3, 8, 18, 53 1, 3, 8, 18, 53 10.33( 13.94) 11.16( 14.10) 10.62( 13.33) 10.07( 13.43) 1 1.5 1, 3, 6, 13, 33 1, 3, 7, 14, 36 1, 3, 6, 13, 35 1, 3, 6, 12, 33 5.83( 6.31) 7.99( 8.79) 6.41( 7.00) 5.46( 5.90) 1 2 1, 2, 4, 7, 18 1, 2, 5, 10, 24 1, 2, 4, 8, 20 1, 2, 4, 7, 16 2.01( 1.92) 1.78( 1.38) 1.86( 1.78) 2.07( 2.19) 2 1 1, 1, 1, 2, 5 1, 1, 1, 2, 4 1, 1, 1, 2, 5 1, 1, 1, 2, 5 2.13( 1.88) 1.99( 1.62) 2.00( 1.72) 2.13( 1.95) 2 1.25 1, 1, 1, 3, 6 1, 1, 1, 2, 5 1, 1, 1, 2, 5 1, 1, 1, 3, 6 2.17( 1.84) 2.15( 1.74) 2.06( 1.69) 2.15( 1.85) 2 1.5 1, 1, 2, 3, 6 1, 1, 2, 3, 5 1, 1, 1, 2, 5 1, 1, 2, 3, 6 2.14( 1.77) 2.35( 1.92) 2.12( 1.67) 2.08( 1.68) 2 2 1, 1, 2, 3, 5 1, 1, 2, 3, 6 1, 1, 2, 3, 5 1, 1, 1, 3, 5 Cauchy(θ, δ) θ δ LPGA LPlog LPLT LPA SL SC MLPA 513.74( 756.02) 499.64( 783.31) 506.50( 725.46) 503.99( 718.57) 499.34( 665.02) 490.63( 695.57) 502.01( 712.87) 0 1 15, 93, 250, 599, 1922 13, 79, 220, 559, 1963 17, 95, 258, 607, 1843 16, 92, 254, 600, 1881 17, 104, 273, 617, 1750 16, 94, 247, 582, 1755 16, 93, 253, 604, 1856 236.68( 377.17) 228.54( 392.99) 218.12( 330.23) 216.61( 325.97) 235.29( 323.43) 220.74( 337.41) 218.75( 330.84) 0 1.25 8, 43, 116, 274, 841 6, 37, 102, 253, 858 7, 42, 110, 259, 787 8, 42, 112, 260, 751 9, 51, 130, 293, 800 7, 42, 111, 260, 791 8, 43, 113, 259, 776 125.36( 198.17) 124.36( 218.89) 111.08( 168.15) 114.81( 179.45) 132.53( 182.93) 116.37( 176.60) 112.07( 172.10) 0 1.5 5, 24, 63, 147, 446 4, 21, 56, 139, 453 4, 23, 59, 135, 385 4, 23, 60, 137, 405 5, 29, 74, 164, 449 4, 23, 60, 139, 407 4, 23, 59, 134, 386 49.77( 80.43) 49.23( 89.84) 43.29( 61.79) 43.39( 64.02) 55.73( 70.76) 45.68( 64.90) 43.02( 59.00) 0 2 2, 10, 26, 59, 172 2, 9, 23, 55, 172 2, 9, 24, 52, 148 2, 9, 24, 52, 148 3, 14, 33, 71, 183 2, 10, 25, 56, 156 2, 10, 24, 53, 149 460.72( 719.07) 428.66( 723.24) 392.95( 634.36) 393.42( 633.48) 378.19( 592.90) 398.26( 644.14) 392.22( 621.28) 0.5 1 11, 73, 208, 525, 1793 10, 60, 176, 467, 1710 10, 60, 172, 446, 1507 10, 61, 172, 449, 1505 10, 61, 172, 438, 1445 10, 61, 176, 449, 1547 10, 61, 175, 447, 1528 216.65( 362.90) 199.54( 363.77) 178.51( 295.74) 178.33( 299.66) 185.54( 288.63) 184.02( 305.26) 178.24( 302.04) 0.5 1.25 6, 35, 99, 246, 805 5, 31, 85, 219, 762 5, 31, 84, 205, 657 6, 31, 83, 201, 660 6, 33, 89, 219, 674 6, 31, 85, 211, 676 5, 30, 83, 203, 657 117.01( 202.49) 112.64( 212.53) 94.95( 158.84) 95.29( 156.09) 105.17( 151.89) 100.76( 161.23) 94.28( 151.88) 0.5 1.5 4, 20, 55, 133, 425 3, 18, 49, 122, 418 3, 18, 47, 110, 345 3, 18, 47, 111, 341 4, 21, 56, 128, 368 3, 19, 50, 118, 366 3, 18, 47, 111, 331 46.69( 74.17) 45.63( 75.70) 38.33( 56.55) 38.64( 56.37) 48.01( 63.20) 41.43( 60.24) 38.54( 59.28) 0.5 2 2, 9, 24, 54, 165 2, 8, 21, 52, 168 2, 8, 20, 46, 134 2, 8, 21, 47, 135 2, 11, 27, 60, 163 2, 9, 22, 50, 144 2, 8, 20, 47, 134 349.14( 648.57) 282.30( 560.25) 206.94( 441.28) 208.01( 456.06) 177.85( 393.05) 225.79( 468.44) 204.63( 441.88) 1 1 6, 38, 122, 359, 1470 5, 31, 97, 281, 1163 4, 22, 67, 198, 843 4, 22, 67, 193, 862 3, 19, 57, 164, 730 4, 23, 73, 217, 967 4, 22, 67, 192, 826 166.00( 336.45) 147.82( 314.90) 104.48( 228.02) 104.96( 226.27) 98.09( 200.88) 113.96( 228.04) 105.35( 220.71) 1 1.25 4, 20, 62, 170, 645 3, 18, 54, 147, 570 3, 14, 39, 104, 397 3, 14, 39, 106, 405 3, 13, 38, 100, 383 3, 15, 44, 117, 450 3, 14, 40, 107, 403 91.38( 179.40) 84.65( 177.55) 61.11( 119.63) 62.90( 122.65) 62.11( 114.09) 67.49( 138.46) 60.93( 123.02) 1 1.5 2, 13, 37, 98, 348 2, 12, 34, 88, 318 2, 10, 27, 66, 227 2, 10, 27, 66, 234 2, 10, 28, 69, 230 2, 10, 29, 73, 250 2, 10, 26, 65, 228 38.85( 66.70) 36.63( 65.88) 28.79( 47.18) 28.89( 45.97) 33.80( 53.22) 31.29( 51.38) 28.34( 45.09) 1 2 2, 7, 18, 44, 142 1, 6, 17, 40, 134 1, 6, 14, 33, 102 1, 6, 14, 33, 104 2, 7, 17, 40, 117 1, 6, 16, 36, 111 1, 6, 14, 33, 101 120.14( 379.52) 91.36( 288.17) 36.31( 149.44) 34.99( 134.48) 25.45( 103.94) 48.19( 202.15) 36.15( 143.84) 2 1 1, 5, 19, 74, 511 1, 6, 20, 68, 377 1, 3, 8, 24, 135 1, 3, 8, 24, 132 1, 3, 6, 17, 91 1, 3, 9, 29, 177 1, 3, 8, 23, 133 65.41( 220.00) 54.05( 158.48) 23.27( 67.55) 23.97( 75.81) 18.23( 59.92) 28.55( 99.41) 24.23( 78.78) 2 1.25 1, 4, 14, 45, 266 1, 5, 15, 44, 211 1, 3, 7, 19, 88 1, 3, 7, 19, 89 1, 3, 6, 15, 65 1, 3, 8, 22, 106 1, 3, 7, 19, 92 39.26( 106.11) 35.89( 96.78) 17.89( 53.25) 18.08( 54.70) 14.91( 44.33) 20.54( 55.12) 17.98( 47.90) 2 1.5 1, 4, 11, 32, 161 1, 4, 12, 32, 137 1, 3, 7, 16, 64 1, 3, 7, 16, 64 1, 3, 6, 14, 52 1, 3, 7, 18, 77 1, 3, 7, 16, 65 19.93( 45.02) 19.96( 51.55) 11.83( 23.97) 11.72( 26.64) 11.76( 21.94) 13.07( 26.74) 11.62( 23.85) 2 2 1, 3, 8, 19, 76 1, 3, 8, 20, 73 1, 2, 6, 13, 41 1, 2, 6, 13, 41 1, 3, 6, 13, 39 1, 3, 6, 13, 47 1, 2, 6, 12, 40 θ

20

ACCEPTED MANUSCRIPT

Table 6: Comparisons of various Shewhart charts under SE(θ, δ) and Gumbel(θ, δ) distributions for m=100, n=5 and ARL0 =500.

1.5

0

2

0.5

1

0.5 1.25 0.5

1.5

0.5

2

1

1

1

1.25

1

1.5

1

2

2

1

2

1.25

2

1.5

2

2

θ

δ

0

1

0

1.25

0

1.5

0

2

0.5

1

0.5 1.25 0.5

1.5

0.5

2

1

1

1

1.25

1

1.5

1

2

2

1

2

1.25

2

1.5

2

2

CR IP T

0

AN US

1.25

M

0

ED

1

PT

0

SE(θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 515.58( 758.92) 498.59( 780.86) 501.25( 719.28) 493.46( 744.69) 505.30( 672.10) 500.98( 720.25) 489.83( 750.98) 15, 92, 252, 612, 1987 13, 79, 223, 554, 1968 16, 94, 250, 597, 1854 14, 85, 234, 570, 1871 18, 105, 277, 627, 1782 16, 92, 250, 601, 1832 14, 79, 219, 545, 1859 181.49( 292.49) 163.79( 280.27) 184.51( 271.97) 169.74( 277.48) 197.20( 284.07) 179.05( 285.80) 161.78( 275.95) 6, 34, 90, 211, 642 5, 29, 76, 184, 591 7, 38, 98, 222, 650 6, 31, 83, 197, 612 7, 40, 105, 240, 685 6, 35, 91, 208, 620 5, 28, 78, 184, 579 61.80( 95.01) 54.82( 88.41) 66.86( 92.31) 57.92( 83.50) 67.94( 92.89) 61.80( 92.65) 54.61( 83.62) 3, 13, 32, 73, 215 2, 11, 28, 65, 191 3, 15, 37, 82, 227 2, 12, 31, 70, 202 3, 15, 38, 85, 230 3, 13, 34, 75, 212 2, 11, 28, 65, 191 14.35( 17.60) 12.51( 15.51) 17.07( 20.44) 14.38( 17.60) 16.82( 19.56) 14.90( 17.18) 13.16( 16.16) 1, 4, 9, 18, 46 1, 3, 8, 16, 40 1, 5, 11, 22, 55 1, 4, 9, 19, 46 1, 5, 11, 22, 54 1, 4, 9, 19, 47 1, 3, 8, 17, 42 310.98( 619.68) 333.46( 673.50) 258.66( 509.40) 289.69( 592.09) 163.11( 301.92) 229.79( 462.08) 295.08( 603.71) 6, 36, 109, 300, 1257 6, 37, 109, 314, 1398 6, 34, 99, 261, 1015 6, 34, 101, 276, 1166 4, 25, 70, 179, 606 5, 30, 86, 230, 896 5, 34, 102, 279, 1211 61.49( 120.04) 58.54( 116.20) 55.10( 92.64) 54.83( 104.70) 43.86( 72.89) 48.66( 82.17) 54.81( 110.65) 2, 10, 28, 67, 224 2, 9, 25, 62, 214 2, 10, 27, 64, 196 2, 10, 25, 60, 198 2, 8, 22, 51, 155 2, 9, 23, 55, 177 2, 9, 25, 60, 196 21.26( 32.69) 19.68( 30.75) 21.50( 30.67) 19.87( 30.38) 17.98( 24.98) 18.51( 25.89) 19.16( 29.21) 1, 5, 11, 25, 73 1, 4, 10, 23, 67 1, 5, 12, 26, 73 1, 4, 11, 24, 68 1, 4, 10, 22, 60 1, 4, 10, 23, 62 1, 4, 10, 23, 66 6.69( 7.70) 5.87( 6.73) 7.22( 8.50) 6.32( 7.26) 6.40( 7.09) 6.18( 7.04) 5.98( 6.97) 1, 2, 4, 8, 21 1, 2, 4, 7, 18 1, 2, 5, 9, 23 1, 2, 4, 8, 19 1, 2, 4, 8, 20 1, 2, 4, 8, 19 1, 2, 4, 7, 18 44.97( 125.01) 37.36( 99.67) 22.87( 52.73) 33.13( 107.58) 16.58( 41.29) 23.02( 69.17) 32.32( 97.81) 1, 6, 16, 42, 169 1, 5, 13, 34, 141 1, 3, 9, 22, 84 1, 4, 11, 30, 120 1, 3, 7, 17, 60 1, 3, 9, 22, 84 1, 4, 11, 30, 121 13.24( 23.08) 10.49( 19.06) 8.17( 13.79) 10.05( 17.84) 6.66( 10.54) 7.95( 12.71) 9.43( 17.14) 1, 3, 6, 14, 47 1, 2, 5, 12, 36 1, 2, 4, 9, 28 1, 2, 5, 11, 35 1, 2, 3, 7, 23 1, 2, 4, 9, 27 1, 2, 5, 10, 33 6.45( 8.93) 5.09( 7.30) 4.49( 5.81) 5.01( 6.97) 3.79( 4.82) 4.33( 5.46) 4.73( 6.20) 1, 2, 4, 8, 21 1, 1, 3, 6, 16 1, 1, 3, 5, 14 1, 1, 3, 6, 16 1, 1, 2, 4, 12 1, 1, 3, 5, 14 1, 1, 3, 6, 15 2.96( 3.00) 2.42( 2.34) 2.40( 2.31) 2.52( 2.50) 2.14( 1.94) 2.30( 2.19) 2.34( 2.20) 1, 1, 2, 4, 8 1, 1, 2, 3, 7 1, 1, 2, 3, 7 1, 1, 2, 3, 7 1, 1, 1, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 1.45( 2.08) 1.51( 1.75) 1.11( 0.58) 1.34( 1.44) 1.06( 0.36) 1.13( 0.69) 1.40( 1.67) 1, 1, 1, 1, 3 1, 1, 1, 1, 4 1, 1, 1, 1, 2 1, 1, 1, 1, 3 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1, 1, 1, 1, 3 1.20( 0.87) 1.23( 0.80) 1.05( 0.32) 1.17( 0.71) 1.03( 0.23) 1.07( 0.40) 1.18( 0.69) 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1.12( 0.57) 1.13( 0.50) 1.03( 0.20) 1.09( 0.45) 1.02( 0.16) 1.04( 0.24) 1.10( 0.43) 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1.05( 0.28) 1.05( 0.25) 1.01( 0.12) 1.04( 0.24) 1.01( 0.09) 1.01( 0.15) 1.04( 0.22) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 Gumbel(θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 509.15( 744.60) 497.81( 776.34) 504.43( 719.89) 503.57( 758.10) 497.70( 661.64) 494.08( 705.81) 487.10( 742.59) 15, 91, 246, 598, 1928 13, 79, 221, 565, 1937 17, 97, 256, 598, 1851 14, 88, 239, 584, 1901 18, 104, 271, 618, 1737 16, 93, 250, 588, 1825 13, 79, 217, 545, 1830 84.72( 109.66) 60.29( 88.07) 85.37( 111.92) 71.48( 99.35) 117.15( 145.15) 87.89( 115.28) 61.94( 89.34) 4, 19, 49, 107, 284 3, 13, 33, 73, 210 4, 20, 49, 107, 284 3, 16, 40, 89, 246 5, 28, 70, 151, 385 4, 20, 51, 111, 294 3, 13, 34, 75, 213 28.64( 33.73) 17.71( 21.69) 28.46( 33.43) 22.61( 28.49) 44.54( 50.55) 29.73( 35.05) 18.48( 22.56) 2, 7, 18, 37, 91 1, 5, 11, 23, 57 2, 7, 18, 37, 91 1, 6, 14, 29, 74 2, 11, 28, 59, 143 2, 8, 18, 39, 95 1, 5, 11, 23, 60 8.40( 8.56) 4.91( 4.76) 8.12( 8.31) 6.53( 7.12) 13.95( 14.79) 8.66( 8.89) 5.20( 5.35) 1, 3, 6, 11, 25 1, 2, 3, 6, 14 1, 3, 5, 11, 24 1, 2, 4, 8, 20 1, 4, 9, 19, 43 1, 3, 6, 12, 26 1, 2, 3, 7, 16 333.52( 638.69) 373.11( 705.68) 293.97( 530.30) 332.17( 640.78) 223.65( 406.11) 269.80( 508.70) 348.04( 662.04) 7, 42, 123, 332, 1343 7, 45, 132, 369, 1549 7, 43, 121, 311, 1129 7, 42, 121, 329, 1341 6, 35, 96, 244, 842 7, 37, 106, 277, 1052 7, 44, 126, 343, 1407 70.29( 119.58) 65.10( 111.82) 70.55( 111.00) 65.66( 112.50) 66.94( 100.95) 64.34( 103.82) 63.19( 111.17) 3, 13, 34, 80, 248 2, 12, 31, 74, 233 3, 14, 37, 84, 247 2, 12, 32, 75, 235 3, 14, 36, 80, 231 3, 13, 33, 75, 223 2, 12, 31, 71, 225 25.54( 34.21) 20.16( 26.78) 26.19( 32.55) 22.54( 29.81) 29.44( 37.58) 24.99( 32.99) 20.35( 27.42) 1, 6, 15, 31, 85 1, 5, 12, 25, 67 2, 7, 16, 33, 86 1, 5, 13, 28, 75 2, 7, 18, 38, 96 1, 6, 15, 31, 82 1, 5, 12, 25, 68 7.97( 8.75) 5.34( 5.54) 8.08( 8.57) 6.62( 7.19) 10.90( 11.48) 8.03( 8.39) 5.53( 5.83) 1, 2, 5, 10, 24 1, 2, 4, 7, 16 1, 2, 5, 11, 24 1, 2, 4, 8, 20 1, 3, 7, 14, 33 1, 2, 5, 11, 24 1, 2, 4, 7, 16 63.04( 153.25) 64.91( 166.71) 45.58( 95.34) 58.31( 142.00) 34.94( 68.54) 42.64( 88.63) 61.11( 162.40) 2, 9, 24, 61, 235 2, 9, 23, 61, 243 2, 8, 20, 49, 163 2, 8, 22, 57, 210 1, 6, 16, 38, 126 2, 7, 19, 45, 155 2, 8, 23, 58, 217 21.34( 35.20) 21.46( 36.22) 20.53( 30.07) 20.49( 32.12) 17.15( 24.68) 18.48( 27.13) 20.53( 33.60) 1, 4, 11, 25, 74 1, 4, 11, 24, 74 1, 5, 11, 25, 68 1, 4, 11, 24, 70 1, 4, 10, 21, 57 1, 4, 10, 22, 62 1, 4, 10, 24, 71 11.45( 15.30) 10.58( 13.89) 12.05( 14.88) 10.95( 14.05) 11.11( 13.21) 10.75( 13.02) 10.52( 13.68) 1, 3, 7, 14, 37 1, 3, 6, 13, 35 1, 3, 7, 15, 39 1, 3, 6, 14, 35 1, 3, 7, 14, 35 1, 3, 7, 14, 34 1, 3, 6, 13, 34 5.41( 5.65) 4.35( 4.43) 5.89( 6.00) 4.92( 5.05) 6.50( 6.80) 5.47( 5.61) 4.49( 4.60) 1, 2, 4, 7, 16 1, 1, 3, 6, 13 1, 2, 4, 8, 17 1, 2, 3, 6, 15 1, 2, 4, 8, 19 1, 2, 4, 7, 16 1, 2, 3, 6, 13 3.33( 5.69) 3.14( 4.60) 2.34( 2.65) 3.01( 4.75) 2.06( 2.09) 2.33( 2.69) 3.01( 4.07) 1, 1, 2, 4, 10 1, 1, 2, 3, 10 1, 1, 1, 3, 7 1, 1, 2, 3, 9 1, 1, 1, 2, 5 1, 1, 1, 3, 6 1, 1, 2, 3, 9 2.73( 3.04) 2.56( 2.71) 2.33( 2.17) 2.52( 2.58) 2.13( 1.87) 2.25( 2.18) 2.52( 2.68) 1, 1, 2, 3, 8 1, 1, 2, 3, 7 1, 1, 2, 3, 6 1, 1, 2, 3, 7 1, 1, 1, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 7 2.45( 2.28) 2.33( 2.19) 2.40( 2.09) 2.36( 2.18) 2.22( 1.88) 2.24( 1.96) 2.29( 2.06) 1, 1, 2, 3, 7 1, 1, 2, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 2.18( 1.73) 2.03( 1.60) 2.38( 1.95) 2.13( 1.71) 2.28( 1.83) 2.14( 1.68) 2.04( 1.61) 1, 1, 2, 3, 6 1, 1, 1, 2, 5 1, 1, 2, 3, 6 1, 1, 2, 3, 5 1, 1, 2, 3, 6 1, 1, 2, 3, 5 1, 1, 1, 2, 5

CE

δ

AC

θ

21

ACCEPTED MANUSCRIPT Table 7: Comparisons of various Shewhart charts under different distributions for m=100, n=5 and ARL0 =500.

0 1

1

0 2

1

1 1

1

1 2

5

0 1

5

0 2

5

1 1

5

1 2

ϑ

θ δ

0.5 0 1 0.5 0 2 0.5 1 1 0.5 1 2 1

0 1

1

0 2

1

1 1

1

1 2

5

0 1

5

0 2

5

1 1

5

1 2

ϑ

θ δ

0.5 0 1 0.5 0 2 0.5 1 1 0.5 1 2 1

0 1

1

0 2

1

1 1

1

1 2

5

0 1

5

0 2

5

1 1

5

1 2

CR IP T

1

AN US

0.5 1 2

M

0.5 1 1

ED

0.5 0 2

PT

0.5 0 1

N (θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 14.24( 17.18) 12.54( 15.52) 17.37( 20.81) 13.83( 17.23) 16.72( 19.38) 14.94( 17.59) 12.72( 15.67) 1, 4, 9, 18, 46 1, 3, 8, 16, 40 1, 5, 11, 22, 56 1, 4, 8, 17, 45 1, 5, 10, 22, 53 1, 4, 9, 19, 48 1, 3, 8, 16, 41 2.22( 1.67) 1.75( 1.18) 2.48( 1.97) 1.99( 1.49) 3.16( 2.68) 2.44( 1.93) 1.76( 1.20) 1, 1, 2, 3, 6 1, 1, 1, 2, 4 1, 1, 2, 3, 6 1, 1, 1, 2, 5 1, 1, 2, 4, 8 1, 1, 2, 3, 6 1, 1, 1, 2, 4 46.02( 61.03) 35.24( 49.03) 47.23( 62.29) 41.14( 56.43) 58.15( 75.90) 46.51( 61.80) 35.31( 50.68) 2, 11, 26, 57, 156 2, 8, 19, 43, 120 2, 11, 27, 59, 156 2, 9, 23, 51, 140 3, 14, 34, 74, 193 2, 11, 27, 58, 155 2, 8, 19, 43, 121 3.57( 3.12) 2.22( 1.72) 3.51( 3.10) 2.89( 2.57) 5.80( 5.47) 3.70( 3.27) 2.25( 1.78) 1, 1, 3, 5, 10 1, 1, 2, 3, 6 1, 1, 3, 5, 10 1, 1, 2, 4, 8 1, 2, 4, 8, 17 1, 1, 3, 5, 10 1, 1, 2, 3, 6 507.01( 729.25) 497.45( 775.22) 497.91( 715.33) 515.50( 777.89) 496.73( 656.78) 495.56( 703.21) 497.75( 787.13) 15, 93, 250, 611, 1843 13, 79, 225, 560, 1925 15, 92, 249, 595, 1830 14, 86, 241, 600, 1948 17, 101, 271, 622, 1743 16, 94, 249, 591, 1854 13, 79, 220, 550, 1967 6.92( 6.91) 3.98( 3.75) 6.69( 6.66) 5.40( 5.55) 11.69( 11.86) 7.14( 7.05) 4.07( 3.88) 1, 2, 5, 9, 20 1, 1, 3, 5, 11 1, 2, 5, 9, 20 1, 2, 4, 7, 16 1, 3, 8, 16, 35 1, 2, 5, 9, 21 1, 1, 3, 5, 12 8.50( 10.96) 8.18( 11.39) 8.60( 10.29) 8.39( 11.19) 7.72( 9.40) 7.73( 9.42) 7.88( 10.02) 1, 2, 5, 10, 28 1, 2, 5, 10, 27 1, 2, 5, 11, 27 1, 2, 5, 10, 27 1, 2, 5, 10, 24 1, 2, 5, 10, 24 1, 2, 5, 10, 25 3.42( 3.02) 2.57( 2.11) 3.87( 3.49) 3.02( 2.70) 4.95( 4.61) 3.78( 3.44) 2.61( 2.14) 1, 1, 2, 4, 9 1, 1, 2, 3, 7 1, 1, 3, 5, 11 1, 1, 2, 4, 8 1, 2, 3, 7, 14 1, 1, 3, 5, 11 1, 1, 2, 3, 7 8.99( 18.63) 8.50( 16.18) 5.91( 9.55) 8.64( 17.25) 4.86( 6.60) 5.85( 9.51) 8.26( 15.56) 1, 2, 4, 10, 31 1, 2, 4, 9, 29 1, 2, 3, 7, 19 1, 2, 4, 9, 30 1, 1, 3, 6, 15 1, 2, 3, 7, 19 1, 2, 4, 9, 29 1.17( 0.46) 1.13( 0.39) 1.24( 0.56) 1.16( 0.45) 1.22( 0.52) 1.18( 0.47) 1.13( 0.39) 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1.02( 0.15) 1.02( 0.13) 1.01( 0.10) 1.02( 0.13) 1.01( 0.08) 1.01( 0.09) 1.01( 0.13) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1.00( 0.06) 1.00( 0.04) 1.01( 0.08) 1.00( 0.06) 1.01( 0.08) 1.00( 0.07) 1.00( 0.04) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 Cauchy(θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 14.23( 17.19) 12.63( 15.69) 16.84( 19.89) 17.11( 20.13) 16.76( 20.11) 14.96( 17.65) 17.31( 20.72) 1, 4, 9, 18, 46 1, 3, 8, 16, 41 1, 5, 11, 22, 53 1, 5, 11, 22, 55 1, 4, 10, 22, 53 1, 4, 9, 19, 48 1, 5, 11, 22, 54 5.05( 5.29) 4.63( 4.94) 5.49( 5.55) 5.44( 5.57) 6.05( 6.10) 5.21( 5.35) 5.44( 5.61) 1, 2, 3, 6, 15 1, 2, 3, 6, 14 1, 2, 4, 7, 16 1, 2, 4, 7, 16 1, 2, 4, 8, 18 1, 2, 4, 7, 15 1, 2, 4, 7, 16 20.09( 23.52) 18.90( 23.33) 28.22( 32.77) 28.14( 32.76) 31.11( 34.59) 25.19( 29.05) 28.08( 32.40) 1, 5, 13, 26, 64 1, 5, 11, 24, 61 2, 7, 18, 37, 90 2, 7, 18, 37, 89 2, 9, 20, 42, 97 2, 7, 16, 33, 80 2, 8, 18, 37, 89 6.36( 6.70) 5.75( 6.39) 7.25( 7.51) 7.32( 7.70) 9.25( 9.76) 7.04( 7.30) 7.24( 7.59) 1, 2, 4, 8, 19 1, 2, 4, 7, 17 1, 2, 5, 10, 22 1, 2, 5, 10, 22 1, 3, 6, 12, 28 1, 2, 5, 9, 21 1, 2, 5, 9, 22 513.74( 756.02) 499.64( 783.31) 506.50( 725.46) 503.99( 718.57) 499.34( 665.02) 490.63( 695.57) 502.01( 712.87) 15, 93, 250, 599, 1922 13, 79, 220, 559, 1963 17, 95, 258, 607, 1843 16, 92, 254, 600, 1881 17, 104, 273, 617, 1750 16, 94, 247, 582, 1755 16, 93, 253, 604, 1856 49.77( 80.43) 49.23( 89.84) 43.29( 61.79) 43.39( 64.02) 55.73( 70.76) 45.68( 64.90) 43.02( 59.00) 2, 10, 26, 59, 172 2, 9, 23, 55, 172 2, 9, 24, 52, 148 2, 9, 24, 52, 148 3, 14, 33, 71, 183 2, 10, 25, 56, 156 2, 10, 24, 53, 149 349.14( 648.57) 282.30( 560.25) 206.94( 441.28) 208.01( 456.06) 177.85( 393.05) 225.79( 468.44) 204.63( 441.88) 6, 38, 122, 359, 1470 5, 31, 97, 281, 1163 4, 22, 67, 198, 843 4, 22, 67, 193, 862 3, 19, 57, 164, 730 4, 23, 73, 217, 967 4, 22, 67, 192, 826 38.85( 66.70) 36.63( 65.88) 28.79( 47.18) 28.89( 45.97) 33.80( 53.22) 31.29( 51.38) 28.34( 45.09) 2, 7, 18, 44, 142 1, 6, 17, 40, 134 1, 6, 14, 33, 102 1, 6, 14, 33, 104 2, 7, 17, 40, 117 1, 6, 16, 36, 111 1, 6, 14, 33, 101 9.03( 24.44) 8.58( 17.60) 5.95( 9.67) 5.97( 8.99) 4.83( 6.50) 5.76( 8.78) 5.94( 9.03) 1, 2, 4, 9, 30 1, 2, 4, 9, 29 1, 2, 3, 7, 20 1, 2, 3, 7, 20 1, 1, 3, 6, 15 1, 1, 3, 7, 19 1, 2, 3, 7, 19 2.09( 2.19) 2.11( 2.47) 1.80( 1.50) 1.81( 1.46) 1.65( 1.23) 1.75( 1.42) 1.81( 1.49) 1, 1, 1, 2, 6 1, 1, 1, 2, 6 1, 1, 1, 2, 4 1, 1, 1, 2, 4 1, 1, 1, 2, 4 1, 1, 1, 2, 4 1, 1, 1, 2, 4 3.70( 8.45) 3.43( 6.32) 1.97( 2.66) 1.97( 2.69) 1.69( 1.91) 2.12( 3.30) 1.97( 2.70) 1, 1, 2, 4, 12 1, 1, 2, 3, 11 1, 1, 1, 2, 5 1, 1, 1, 2, 5 1, 1, 1, 2, 4 1, 1, 1, 2, 6 1, 1, 1, 2, 5 1.41( 1.20) 1.44( 1.36) 1.21( 0.67) 1.21( 0.62) 1.15( 0.49) 1.22( 0.69) 1.21( 0.62) 1, 1, 1, 1, 3 1, 1, 1, 1, 3 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 SE(θ, δ) LPGA LPlog LPLT LPA SL SC MLPA 14.03( 16.73) 12.53( 15.16) 17.19( 20.05) 14.44( 17.42) 16.86( 19.74) 15.07( 17.91) 13.44( 16.74) 1, 4, 9, 18, 44 1, 3, 8, 16, 40 1, 5, 11, 22, 55 1, 4, 9, 18, 47 1, 5, 11, 22, 54 1, 4, 9, 19, 48 1, 3, 8, 17, 43 21.64( 25.03) 13.26( 15.73) 21.19( 24.74) 17.66( 21.29) 33.67( 38.10) 22.11( 24.98) 14.55( 17.70) 1, 6, 14, 28, 69 1, 4, 8, 17, 42 1, 6, 13, 28, 66 1, 5, 11, 23, 57 2, 9, 22, 45, 105 1, 6, 14, 29, 71 1, 4, 9, 19, 47 485.57( 879.37) 398.81( 798.44) 303.49( 629.32) 368.02( 730.78) 201.61( 439.15) 287.05( 615.56) 363.70( 745.97) 8, 51, 162, 479, 2215 6, 37, 120, 366, 1786 5, 29, 94, 282, 1293 6, 36, 113, 347, 1617 4, 21, 66, 190, 811 5, 28, 88, 257, 1228 5, 33, 106, 326, 1617 16.68( 23.41) 12.06( 17.53) 14.30( 19.65) 13.49( 19.30) 12.06( 17.16) 12.48( 17.22) 12.00( 17.11) 1, 4, 9, 20, 56 1, 3, 7, 15, 40 1, 3, 8, 18, 49 1, 3, 7, 16, 45 1, 3, 7, 15, 40 1, 3, 7, 15, 42 1, 3, 7, 14, 41 515.58( 758.92) 498.59( 780.86) 501.25( 719.28) 493.46( 744.69) 505.30( 672.10) 500.98( 720.25) 489.83( 750.98) 15, 92, 252, 612, 1987 13, 79, 223, 554, 1968 16, 94, 250, 597, 1854 14, 85, 234, 570, 1871 18, 105, 277, 627, 1782 16, 92, 250, 601, 1832 14, 79, 219, 545, 1859 14.35( 17.60) 12.51( 15.51) 17.07( 20.44) 14.38( 17.60) 16.82( 19.56) 14.90( 17.18) 13.16( 16.16) 1, 4, 9, 18, 46 1, 3, 8, 16, 40 1, 5, 11, 22, 55 1, 4, 9, 19, 46 1, 5, 11, 22, 54 1, 4, 9, 19, 47 1, 3, 8, 17, 42 44.97( 125.01) 37.36( 99.67) 22.87( 52.73) 33.13( 107.58) 16.58( 41.29) 23.02( 69.17) 32.32( 97.81) 1, 6, 16, 42, 169 1, 5, 13, 34, 141 1, 3, 9, 22, 84 1, 4, 11, 30, 120 1, 3, 7, 17, 60 1, 3, 9, 22, 84 1, 4, 11, 30, 121 2.96( 3.00) 2.42( 2.34) 2.40( 2.31) 2.52( 2.50) 2.14( 1.94) 2.30( 2.19) 2.34( 2.20) 1, 1, 2, 4, 8 1, 1, 2, 3, 7 1, 1, 2, 3, 7 1, 1, 2, 3, 7 1, 1, 1, 3, 6 1, 1, 2, 3, 6 1, 1, 2, 3, 6 8.86( 16.93) 8.56( 15.82) 6.02( 9.40) 7.57( 13.82) 4.87( 6.60) 5.84( 8.96) 7.70( 18.07) 1, 2, 4, 10, 30 1, 2, 4, 9, 30 1, 2, 3, 7, 20 1, 2, 4, 8, 26 1, 1, 3, 6, 16 1, 2, 3, 7, 19 1, 2, 4, 8, 26 1.07( 0.29) 1.05( 0.23) 1.05( 0.25) 1.06( 0.25) 1.05( 0.23) 1.04( 0.22) 1.05( 0.23) 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1.21( 0.80) 1.21( 0.74) 1.07( 0.31) 1.16( 0.65) 1.04( 0.25) 1.08( 0.37) 1.17( 0.61) 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1, 1, 1, 1, 1 1, 1, 1, 1, 2 1, 1, 1, 1, 2 1.00( 0.03) 1.00( 0.02) 1.00( 0.01) 1.00( 0.02) 1.00( 0.01) 1.00( 0.01) 1.00( 0.02) 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1 1, 1, 1, 1, 1

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Real data applications

It is important to discuss the implementation strategies of the proposed schemes with real data, from a practical point of view. In this section, we use two examples from Montgomery (2009) to illustrate the comparisons of the proposed procedures. Our first example is based on the familiar piston ring data. We consider another example of monitoring compressive strength of parts manufactured by an injection molding process. Example 1. Monitoring inside diameters of forged automobile engine piston rings:

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The goal is to establish a statistical control of the inside diameters of the rings manufactured by this process. Table 6.3 of Montgomery (2009) presents 25 samples, and each sample consists of 5 piston rings. We use different approaches of distribution-free Phase-I analysis applying various codes available in the R package “dfphase1”. See Capizzi and Masarotto (2018) for more details. The results provide no indication of an OOC signal. Therefore, we may consider these 125 samples as a set of reference sample, and consequently, the reference sample size is m = 125. Table 6E.7 of Montgomery (2009) contains another 15 samples each

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of size 5, related to inside diameters of piston rings. Clearly, the test sample size is n = 5. We estimate ˆ based on the 125 reference sample observations by using Equation (2). We see that the the tail-weight W ˆ = 1.66. The Bowley’s skewness based on 125 reference sample observations is 0.0769. These findings W indicate that the underlying process distribution is likely to be symmetric and has a medium tail-weight. Naturally, we should choose the LPlog statistic for the adaptive LPA and MLPA schemes. In this example, the LPlog, LPA and MLPA schemes will lead to the same conclusion. All the monitoring schemes are calibrated so that they have ARL0 values of 500 for a fair comparison. We find the U CL H values for

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the LPGA, LPlog (LPA and MLPA), LPLT, SL and SC schemes via simulation based on Monte-Carlo. The U CL values are given in Table 8. In the same table, we also display the plotting statistics of various monitoring schemes. The values of the plotting statistics corresponding to the samples that produce the

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OOC signals are indicated with dark gray shades. From Table 8, we see that all the monitoring schemes considered in this context, conform no change in the process distribution for the first 11 test samples. These NSPM schemes further indicate that the process is OOC for the first time at sample number 12, and OOC

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signal persists from sample number 12 onwards until sample number 14. Example 2. Monitoring compressive strengths of parts manufactured by an injection mold-

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ing process:

We illustrate the proposed schemes with another data related to the parts manufactured by an injection molding process that underwent a compressive strength test. Twenty samples of five parts each are collected,

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and the compressive strengths are shown in Table 6E.11 of Montgomery (2009). As in previous example, we apply various tools for distribution-free Phase-I analysis on this data. The results show that no point falls outside the control limits. Therefore, we may safely conclude that the process is IC. We consider these 100 observations as a set of the reference sample. Thus, in this context, m = 100. Fifteen new samples of five parts each are collected and the compressive strengths are shown in Table 6E.12 of Montgomery ˆ of tail-weight and the Bowley’s skewness (2009). Hence the test sample size is: n = 5. The estimate W based on 100 reference sample observations are 1.546 and 0.0146, respectively. Therefore, the underlying process distribution is likely to be symmetric and short-tailed. Naturally, we apply the LPGA statistic on this data example for LPA scheme, whereas, we select the LPlog statistic for MLPA scheme. That is, for this data set, the LPGA and LPA schemes will offer the same results, while the MLPA scheme will

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be equivalent to the LPlog scheme. Next, we consider this data set to illustrate the effectiveness of the aforementioned schemes. As in previous example, for a fair comparison, all the monitoring schemes are calibrated to ensure that the ARL0 values of all schemes equal 500. The U CL, H, of various schemes are enumerated by simulation and are presented in Table 9. In the same table, we also present 15 plotting statistics corresponding to each scheme. We use the dark gray shades to indicate the OOC signals. We further display the observed values of 15 plotting statistics for the LPGA (LPA), LPlog (MLPA), LPLT, SL and SC schemes in Figs. 2-6, respectively. From Table 9 and Figs. 2-6, we find OOC signals at the 4th, 6th and 11th test samples from the LPGA scheme. The LPlog scheme produces OOC signals at the 4th, 7th

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and 11th test samples and the LPLT scheme offers a signal only at sample 6. There is no indication of an OOC condition in the SL and SC schemes. We can see certain benefit of the adaptive monitoring schemes based on additional information on symmetry and tail-weight of the underlying process. Clearly, the LPA (LPGA) and MLPA (LPlog) schemes can detect shifts more quickly compared to their competitors for this data set. Our proposed adaptive schemes minimize the loss of information and offer enhanced performance compared to the traditional (non-adaptive) NSPM schemes. We expect that the practitioners will regard

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this as a very useful tool.

Table 8: Plotting statistics for piston ring data.

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SERIAL LPGA LPlog, LPA, MLPA LPLT SL SC NO H = 14.49 H = 13.50 H = 11.05 H = 11.40 H = 6.15 1 7.318 6.735 4.736 3.901 2.688 2 0.645 0.120 0.127 0.182 0.112 3 3.721 3.779 3.408 3.776 1.925 4 0.439 1.011 1.438 0.612 0.388 5 1.211 1.787 2.226 3.853 1.150 6 3.594 2.010 1.092 1.733 0.943 7 1.194 1.328 1.332 1.370 0.691 8 1.529 2.270 2.801 3.011 1.332 9 9.873 6.551 3.916 4.451 2.829 10 7.138 7.190 4.784 5.309 2.941 11 1.251 0.773 0.373 0.334 0.262 12 21.932 18.003 13.575 14.639 8.333 13 23.769 25.919 17.114 18.188 10.153 14 33.156 30.209 20.673 21.741 12.593 15 6.865 6.812 4.778 5.225 2.912

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Table 9: Plotting statistics for injection molding process data. SERIAL LPGA, LPA LPlog, MLPA LPLT SL SC NO H = 13.821 H = 12.96 H = 10.85 H = 11.25 H = 5.981 1 1.302 0.024 0.184 0.024 0.038 2 5.987 5.187 4.240 2.962 2.128 3 1.226 1.119 1.089 1.098 0.611 4 16.342 13.158 9.105 10.845 5.879 5 5.537 8.986 4.913 3.008 2.524 6 13.861 12.448 11.105 9.202 5.845 7 10.542 15.537 8.444 6.838 4.518 8 6.377 5.150 5.518 5.295 2.939 9 1.401 4.731 6.123 4.755 2.321 10 3.635 1.765 0.605 0.821 0.451 11 14.631 14.500 10.700 9.166 5.797 12 3.286 1.491 0.777 0.993 0.687 13 0.703 2.424 0.805 0.555 0.401 14 5.034 7.331 4.489 5.350 2.691 15 5.480 7.745 7.341 6.999 3.556

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Figure 2: Phase-II LPGA (LPA) monitoring scheme for injection molding process data.

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Figure 3: Phase-II LPlog (MLPA) monitoring scheme for injection molding process data.

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Figure 4: Phase-II LPLT monitoring scheme for injection molding process data.

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Figure 5: Phase-II SL monitoring scheme for injection molding process data.

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Figure 6: Phase-II SC monitoring scheme for injection molding process data.

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5

Conclusions and considerations

In this paper, we introduce three distribution-free SL type monitoring schemes based on the LPGA, LPlog and LPLT statistics designed for the case of short, medium and long-tailed symmetric distributions, respectively. Further, we use these three statistics to design two adaptive SL type monitoring schemes, one is the LPA scheme which is based on an adaptive test proposed by K¨ossler (2006), and the other is a new scheme with finite sample correction, referred to as the MLPA scheme. Both the IC and the OOC performance of the proposed class of schemes are studied in terms of various characteristics of the run length distribution,

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viz. the mean, the standard deviation and some percentiles. Monitoring schemes proposed here are exactly distribution-free and therefore, the IC performance of the proposed class of schemes remains the same for all univariate continuous distributions. A performance comparison of our proposed schemes is done with the familiar SL and SC schemes for a large class of location-scale models by an extensive simulation study. The comparative study reveals that no individual monitoring scheme is the best in all cases. Nevertheless, the overall performance of our proposed MLPA scheme is similar or very close to the best available scheme for a broad class of location-scale models. B¨ uning and Thadewald (2000) noted that the idea of an

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adaptive scheme is based on selecting an appropriate test for a given data set where the data generating process is unknown. The choice may be made in different ways. Our proposed MLPA scheme is effective and computationally simple. Therefore, we conclude that the MLPA monitoring scheme is one of the best choice when the process distribution is unknown. In future, the authors wish to study further enhancements and refinements, using EWMA and CUSUM statistics. We recommend further research on skewed process

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distributions, especially, for process distributions with shape or skewness parameters.

Appendix A. Proof of Theorem 1

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It is easy to see that

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 P rob RLLP [S] = r = E P rob T1,LP [S] ≤ HLP [S] , ..., Tr−1,LP [S] ≤ HLP [S] , Tr,LP [S] > HLP [S] |Xm .

Note that, given the Phase-I sample Xm , Tj,LP [S] , j = 1, 2, . . . , are conditionally mutually independent and are also identically distributed. Consequently, conditional run-length distribution is geometric. Therefore,

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r−1  i    P rob T1,LP [S] > HLP [S] |Xm P rob RLLP [S] = r = E P rob T1,LP [S] ≤ HLP [S] |Xm  
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r−1 = E P rob T1,LP [S] ≤ HLP [S] |Xm − E P rob T1,LP [S] ≤ HLP [S] |Xm .

Hence the result follows. Acknowledgements

Authors are grateful to the Editor, Prof. Steffen Rebennack, and three anonymous reviewers for their thought provoking comments and numerous constructive suggestions that help us in improving the paper to a great extent.

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References [1] Bakir, S.T. (2006). Distribution-free quality control charts based on signed-rank-like statistics. Communictions in StatisticsTheory and Methods, 35(4), 743-757. [2] Bersimis, S., Koutras, M. B., & Maravelakis, P. E. (2014). A compound control chart for monitoring and controlling high quality processes. European Journal of Operational Research, 233(3), 595-603. [3] B¨ uning, H., & K¨ ossler, W. (1999). The asymptotic power of Jonckheere-type tests for ordered alternatives. Australian & New Zealand Journal of Statistics, 41(1), 67-77. [4] B¨ uning, H., & Thadewald, T. (2000). An adaptive two-sample location-scale test of lepage type for symmetric distributions. Journal of Statistical Computation and Simulation, 65(1-4), 287-310.

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[5] Capizzi, G., & Masarotto, G. (2016). Efficient control chart calibration by simulated stochastic approximation. IIE Transactions, 48(1), 57-65. [6] Capizzi, G., & Masarotto, G. (2018). Phase I distribution-free analysis with the R package dfphase1. Frontiers in Statistical Quality Control 12. Cham: Springer. [7] Celano, G., Castagliola, P., & Chakraborti, S. (2016). Joint Shewhart control charts for location and scale monitoring in finite horizon processes. Computers & Industrial Engineering, 101, 427-439. [8] Chakraborti, S., Qiu, P., & Mukherjee, A. (2015). Editorial to the special issue: nonparametric statistical process control charts. Quality and Reliability Engineering International, 31(1):1-2. [9] Chong, Z. L., Mukherjee, A., & Khoo, M. B. C. (2017). Distribution-free Shewhart-Lepage type premier control schemes for simultaneous monitoring of location and scale. Computers & Industrial Engineering, 104, 201-215.

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