Properties and enhancements of robust likelihood CUSUM control chart

Accepted Manuscript Properties and enhancements of robust likelihood CUSUM control chart Chunjie Wu, Miaomiao Yu, Fang Zhuang PII: DOI: Reference:

S0360-8352(17)30473-4 https://doi.org/10.1016/j.cie.2017.10.005 CAIE 4942

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

30 March 2017 17 August 2017 4 October 2017

Please cite this article as: Wu, C., Yu, M., Zhuang, F., Properties and enhancements of robust likelihood CUSUM control chart, Computers & Industrial Engineering (2017), doi: https://doi.org/10.1016/j.cie.2017.10.005

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

• Chunjie Wu∗ School of Statistics and Management, Shanghai University of Finance and Economics, 777 Guoding Rd., Shanghai, China. E-mail: [email protected] Phone: 86-21-65901084 • Miaomiao Yu School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China. • Fang Zhuang School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China.

1

Properties and enhancements of robust likelihood CUSUM control chart

Abstract: In practice, the robust-likelihood cumulative sum (RLCUSUM) control charts, using truncation of the log-likelihood underpinnings of the CUSUM to ensure robust performance, which perform well to detect small shifts in the mean of heavy-tailed processes and have the ability to discount outliers. In this article, we evaluate the properties of RLCUSUM control schemes used to monitor the mean of contaminated normally distributed processes. An optimal design procedure for RLCUSUM control schemes is given, in addition, several enhancements to RLCUSUM are considered. These include a fast initial response feature that makes the RLCUSUM control schemes more sensitive to start-up cases, three types of truncation of RLCUSUM, which are cut off by horizontal ray, horizontal segment and oblique line respectively against both large and small shifts in a non-normality process. An extensive comparison with the conventional CUSUM and generalized likelihood ratio CUSUM (GCUSUM) in terms of average run length (ARL), shows that the RLCUSUM is more robust than the conventional CUSUM and GCUSUM control charts for its ARLs are monotonous against the process mean. Especially, the oblique line truncated RLCUSUM control chart has the best overall performance in the various cases of shifts. A final application based on monitoring packing 250-gram containers is used to illustrate the implementation of RLCUSUM control schemes. Keywords: Robust-likelihood CUSUM control chart; Contaminated; Average run length; Truncation; Fast initial response

1

Introduction

Page (1954) introduced a cumulative sum (CUSUM) control chart, which used the principles of sequential probability analysis to accumulate the sample information, in order to magnify the effects, thereby enhancing the sensitivity of monitoring small shifts in the production process. However, CUSUM control charts are not effective for detecting large shifts. And the design of CUSUM is usually based on normal distributions. While data in production processes follow heavy-tailed distributions, CUSUM control chart often gives a false alarm for outliers, leading to reduce the efficiency of the production, which means CUSUM control charts are lack of robustness under heavy-tailed distributions. Since the CUSUM control chart was proposed, studies on CUSUM, designed strategies and enhancements have been developed to improve the robustness of the CUSUM control chart. To improve the performances of CUSUM control charts in large shifts, the combined 2

Shewhart-CUSUM quality control schemes were considered (Westgard et al. (1977), Lucas (1982) and Wu et al. (2008)), which gave an out-of-control signal if the most recent samples were outside of Shewhart control limits or if a CUSUM signal was given. Lucas and Crosier (1982b) studied four modified CUSUM control schemes for robustness and found control scheme that ignored the first suspected outlier, but gave an out-of-control signal for two successive outliers, performs well. Siegmund and Venkatraman (1995) studied sequential of a change-point using the generalized likelihood ratio (GLR) statistic and gave approximations to the average run lengths for the special case of detecting a change in a normal mean with known variance. To improve the performances of CUSUM control charts in non-normal distributions, Chatterjee and Qiu (2009) proposed a sequence of bootstrap-based control limits for the CUSUM control charts to eliminate the limitation of normality assumption, which was distribution-free and robust. Reynolds and Stoumbos (2010) considered the problem of obtaining robust control charts for detecting changes in the mean and standard deviation of continuous processes, and showed the performance of two CUSUM chart combinations that could be made to be robust to non-normality. Yang et al. (2010) used a robust location estimator (i.e., the sample median) with the CUSUM control scheme and Nazir et al. (2013) extended this approach by comparing the performance of different CUSUM control charts for phase II monitoring of location, based on mean, median, midrange, Hodges-Lehmann, and trimean statistics under different normal, contaminated normal, and special cause contaminated parent scenarios. More literatures about improving the robustness of CUSUM by nonparametric methods can be referred in Hackl and Maderbacher (1999), Li et al. (2010), Yang and Cheng (2011) and so on. Some researchers also considered to improve the robustness of CUSUM by winsorizing or truncating methods, which could improve the performances of CUSUM control charts in large shifts and non-normal distributions, simultaneously. Hawkins (1993) proposed the robustification of CUSUM for individual observations by winsorizing. Truncation maybe reduce the sensitivity of the CUSUM control chart in the normal distribution, while it has the ability to discount outliers in the heavy-tailed cases, thereby reducing the false alarm rate of control charts. MacEachern et al. (2007) considered truncating the non-monotonic log-likelihood function of a variate under t distribution in order that the log-likelihood ratio becomes monotonous and proposed the robust log-likelihood ratio (RLLR). Relying on the Neyman-Pearson lemma, the Neyman-Pearson test is uniformly most powerful (UMP) for distributions with monotone likelihood ratio. Based on RLLR, MacEachern et al. (2007) proposed robust log-likelihood ratio cumulative sum (RLCUSUM) control scheme, which has a robust performance in monitoring heavy-tailed distributed processes and improves the efficiency on detecting large shifts. Liu et al. (2015) adjusted the log-likelihood ratio function to be a strictly monotonic function by using the tangent line method to ensure the in-control average run length (ARL0 ), under different control limits, to be continuous. And based on the adjusted LLR, a simplified multiple control chart is constructed to detect range shifts in the in-control process. In this paper, motivated by the truncating methods of RLLR, a further study of the oblique line truncated RLCUSUM control chart has been done, which is based on contaminated normal distribution. We evaluate the properties of RLCUSUM control schemes. We propose an optimal design procedure for RLCUSUM control schemes and consider several enhancements to RLCUSUM. A fast initial response feature is showed to make the 3

RLCUSUM control schemes more sensitive to start-up cases. Three truncated types of RLCUSUM are presented, which are cut off by horizontal ray, horizontal segment and oblique line respectively against both large and small shifts in a non-normality process. Then, extensive comparisons with the conventional CUSUM and generalized likelihood ratio CUSUM (GCUSUM) in terms of average run length are presented. Those comparative results show that the RLCUSUM is much better than the conventional CUSUM, especially for large shifts. GCUSUM control chart don’t have a robust performance in detecting the heavy-tailed distributions. For small shifts, the two charts are essentially equivalent. Finally, an application based on monitoring packing 250-gram containers is used to illustrate the implementation of RLCUSUM control schemes. The remainder of the article is organized as follows. In section 2, we introduce basic principles of the CUSUM control chart, including the the sequential probability ratio test (SPRT). Section 3 details the RLLR and enhancements of RLCUSUM control charts based on the RLLR. An optimal design of oblique line truncated RLCUSUM control chart is proposed and extensive comparisons with the conventional CUSUM and GCUSUM and a fast initial response feature are included in Section 4. Section 5 provides an application to illustrate the implementation of RLCUSUM control schemes. Conclusions are provided in Section 6.

2

CUSUM and contaminated normal distribution

In this section, the CUSUM control chart, which is constructed based on the SPRT, and the likelihood ratio of contaminated normal distribution are briefly explained.

2.1

SPRT and CUSUM control chart

The hypothesis of sequential probability ratio test is iid

iid

H0 : x1 , x2 , . . . ∼ f0 (x|θ0 ) v.s. H1 : x1 , x2 , . . . ∼ f1 (x|θ1 ). where the θi , i =0, 1 are the parameters of distribution. Assume that the likelihood ratio n f1 (xi |θ1 ) . The log-likelihood ratio function is function is λn = i=1 n f0 (xi |θ0 ) i=1

ln λn =

n  i=1

f1 (xi |θ1 )  = ln zi , f0 (xi |θ0 ) i=1 n

(1)

i |θ1 ) . Given constants A and B, −∞ < A < 1 < B < +∞, sample where zi = ln ff10 (x (xi |θ0 ) successively. If λn ≥ B, then stop sampling and reject the null hypothesis. If λn ≤ A, then stop sampling and accept the null hypothesis. If A < λ < B, continue sampling. This test is called sequential probability ratio test, written as S(A, B). Suppose the observations x1 , x2 , . . . , xn are independent samples of quality characteristic X, with its probability density function (pdf) denoted as f0 (xi |θ0 ). Then it is needed to test whether a mean shift occurs after time ν, where ν is a nonnegative integer. In other words, the pdfs of xν+1 , xν+2 , . . . , xn change to f1 (xi |θ1 ), i = ν + 1, ν + 2, . . . , n, after time ν, where the parameters of f0 (xi |θ0 ) and f1 (xi |θ1 ) are known. This hypothesis testing problem

4

is equivalent to H0 :xi ∼ f0 (x|θ0 ), i = 1, 2, . . . , n, H1 :∃ν(ν < n), xi ∼ f1 (x|θ1 ), i = ν + 1, ν + 2, . . . , n,  f0 (x|θ0 ), θ ∈ Θ0 Assume that Θ0 = {θ0 }, Θ1 = {θ1 }, Θ = Θ0 ∪ Θ1 and f (x|θ) = f1 (x|θ1 ), θ ∈ Θ1 the likelihood ratio function is

, and

⎧  ν   n  n i |θ0 ) i=1 f0 (x i=ν+1 f1 (xi |θ1 ) i=ν+1 f1 (xi |θ1 ) ⎨   = max , ν < n, max n n supθ∈Θ f (x|θ) i=1 f0 (xi |θ0 ) i=ν+1 f0 (xi |θ0 ) 1≤ν
f (x |θ ) ⎨ max ln 1 i 1 , ν < n, (3) Sn = ln λn = 1≤ν
 θˆ1 = arg max log f1 (xi |θ1 ) . (4)

i=1

And GCUSUM control chart is

n  ⎧

fˆ1 (xi |θˆ1 ) ⎨ max ˆ n = 1≤ν
2.2

(5)

Log-likelihood ratio functions based on the contaminated normal distribution

Suppose that most of the observations x1 , x2 , . . . , xn come from normal distribution N(μ, σ02 ) and few come from N(μ, σ 2 ), which are viewed as interference values. Let , 0 <  < 1, be the degree of contamination in all observations. As n is large, the overall distribution function can be regarded as F (x) = P (X ≤ x) = (1 − )Φ(

5

x−μ x−μ ). ) + Φ( σ0 σ

(6)

where Φ(·) is the cumulative distribution function (cdf) of standard normal distribution. This is called contaminated normal distribution, denoted as X ∼ (1 − )N(μ, σ0 ) + N(μ, σ). Therefore, pdf is  x−μ (1 − ) x − μ ) (7) φ( ) + φ( f (x) = σ0 σ0 σ σ where φ(·) is the pdf of standard normal distribution. Suppose that X comes from contaminated normal distribution and we make the hypothesis that H0 : μ = μ 0

v.s. H1 : μ = μ1 .

where μ1 = μ0 and μ0 is the target value of mean when the process is in control. The according log-likelihood ratio is 2

2

1) 1) (1 − ) σ10 exp(− (x−μ ) +  σ1 exp(− (x−μ ) f (x|μ1 ) 2σ2 2σ02 = ln . zi = ln 2 2 0) 0) f (x|μ0 ) (1 − ) σ10 exp(− (x−μ ) +  σ1 exp(− (x−μ ) 2σ2 2σ2

(8)

0

Given σ = 3σ0 (cf., Wu et al.(2002)), μ0 = 0 and σ0 = 1, the influences of μ1 and  on a log-likelihood ratio are as displayed in Figure 1-Figure 2. Figure 1 shows that the smaller  is, the more volatile the non-monotone interval of LLR is. And from Figure 2, we observe that: 1. LLR of contaminated normal distribution is non-monotonic and exists four extreme point. 2. When μ1 is positive that is under the upward shift case, LLR shows a trend of increase. When μ1 is negative, LLR shows a trend of decreasing. And since the upward shift and the downward shift are symmetric, this paper studies upward one-sided control 1 1 1 + ARL = ARL where chart. Furthermore, for two-sided control chart, we have ARL u t d ARLu , ARLd and ARLt mean upward shift’s, downward shift’s and two-sided shift’s ARL respectively. 3. The non-monotone intervals of LLR grow wider, slope becomes greater and the center moves to the right with the increase of μ1 (μ1 > 0). 0 +μ1 −x|μ1 ) 1) 4. It can be proved that ln ff (μ = − ln ff (x|μ , and the center of LLR is ((μ0 + (μ0 +μ1 −x|μ0 ) (x|μ0 ) μ1 )/2, 0).

6

Pf. f (μ0 + μ1 − x|μ1 ) ln = ln f (μ0 + μ1 − x|μ0 ) = ln

= ln

= ln = ln

(1−) 1 φ( μ0 +μ1σ−x−μ ) + σ φ( μ0 +μ1σ−x−μ1 ) σ0 0 (1−) 0 φ( μ0 +μ1σ−x−μ ) + σ φ( μ0 +μ1σ−x−μ0 ) σ0 0 (1−) φ( μ0σ−x ) + σ φ( μ0σ−x ) σ0 0 (1−) φ( μ1σ−x ) + σ φ( μ1σ−x ) σ0 0 0 0 − (1−) φ( x−μ ) − σ φ( x−μ ) σ0 σ0 σ 1 1 − (1−) φ( x−μ ) − σ φ( x−μ ) σ0 σ0 σ (1−) 0 0 φ( x−μ ) + σ φ( x−μ ) σ0 σ0 σ (1−) x−μ1 x−μ1  φ( σ0 ) + σ φ( σ ) σ0

f (x|μ1 ) f (x|μ0) = − ln f (x|μ1) f (x|μ0 )

Insert Figure 1 about here. Insert Figure 2 about here.

3

Three truncated cases for RLCUSUM control charts

In this section, the properties of robust log-likelihood ratio are presented. Two truncation methods, horizontal ray and horizontal segment truncation, to construct the RLCUSUM control chart are reviewed. We evaluate the properties of RLCUSUM control schemes used to monitor the mean of contaminated normally distributed processes and conclude their cons and pros. At the end of this section, the design of oblique line truncated RLCUSUM control chart is proposed. According to the Neyman-Pearson Lemma, for the one-sided simple hypothesis, if the loglikelihood ratio function is monotonous (MLR), the likelihood ratio test for simple hypothesis, constructed based on the MLR, is a uniformly most powerful test (UMPT). CUSUM control chart is constructed based on the likelihood ratio. However, the LLR of contaminated normal distribution is not monotone, which don’t satisfy the requirement of UMPT. Thus, the improvement of LLR is needed.

3.1

Robust log-likelihood ratio

The RLCUSUM chart is based on a robust log-likelihood ratio (RLLR). This ratio is required to have several properties (cf. MacEachern et al. (2007)): • l∗ should be the log-likelihood ratio for some random variable X. • l∗ should be continuous in X. • l∗ should be stochastically increasing in μ if X comes from some location family with location parameter μ.

7

3.2

Horizontal ray truncated RLCUSUM1 control chart

To construct the RLLR1 (cf. MacEachern et al. (2007)), we create a new variate, censoring small values of X at some point r1 and large values of X at some point r2 . Assume that f (·|μi) and F (·|μi) are used to represent the density and cumulative distribution function respectively for i = 0, 1. We get the RLLR1 as follows ⎧ F (r1 |μ1 ) x ≤ r1 , ⎪ ⎨ln F (r1 |μ0 ) , f (x|μ1 ) ∗ r 1 < x < r2 , (9) l (x) = ln f (x|μ0 ) , ⎪ ⎩ 1−F (r2 |μ1 ) ln 1−F (r2 |μ0 ) , x ≥ r2 To maintain the monotonicity and continuity, r1 and r2 should satisfy F (r1 |μ1 ) f (r1 |μ1 ) = ln , F (r1 |μ0 ) f (r1 |μ0 ) 1 − F (r2 |μ1 ) f (r2 |μ1 ) ln = ln , 1 − F (r2 |μ0 ) f (r2 |μ0 ) ln

(10) (11)

where cut off points (r1 , l∗ (r1 )) and (r2 , l∗ (r2 )) should be between the second and third extreme points of LLR. Based on the above RLLR1 , RLCUSUM1 control chart is built S0∗ = 0,

∗ Sn∗ = max{0, Sn−1 + l∗ (xn )}.

(12)

RLCUSUM1 control chart, generally, is better than the CUSUM control chart. However, not all the LLR functions can find cut-off points, which satisfy the request of RLLR1 under 1 |μ1 ) 1 |μ1 ) heavy-tailed distributions. Assume that g(x) := ln ff (r − ln FF (r and r is the solution (r1 |μ0 ) (r1 |μ0 ) of g(x) = 0, satisfying that r ∈ [x2 , x3 ], where (x2 , y2 ), (x3 , y3 ) are respectively the second and the third extreme points of LLR. In the most cases, there is no such a point r in the field of real numbers, meaning that the RLCUSUM1 control chart can not be constructed in the real. From Figure 3-Figure 4, we can see that the larger values of  and μ1 are and the smaller value of σ is, there is more likely to be no solution of Eq.(10) and Eq.(11). On the other hand, the value of LLR tends to be limitless and the tail information is lost. Insert Figure 3 about here. Insert Figure 4 about here.

3.3

Horizontal segment truncated RLCUSUM2 control chart

Wu et al. (2017) proposed that we can cut off the LLR by the horizontal segment to retain the tail information and improve the performances of RLCUSUM charts when the mean of process is large. The RLLR2 is ⎧ F (p2 |μ1 )−F (p1 |μ1 ) ⎪ ⎨ln F (p2 |μ0 )−F (p1 |μ0 ) , p1 ≤ x ≤ p2 , 4 |μ1 )−F (p3 |μ1 ) (13) , p3 ≤ x ≤ p4 , l# (x) = ln FF (p (p 4 |μ0 )−F (p3 |μ0 ) ⎪ ⎩ f (x|μ1 ) Others. ln f (x|μ0 ) , 8

where p1 , p2 , p3 , p4 should satisfy f (p1 |μ1 ) f (p2 |μ1 ) F (p2 |μ1 ) − F (p1 |μ1 ) = ln = ln , F (p2 |μ0 ) − F (p1 |μ0 ) f (p1 |μ0 ) f (p2 |μ0 ) F (p4 |μ1 ) − F (p3 |μ1 ) f (p3 |μ1 ) f (p4 |μ1 ) ln = ln = ln . F (p4 |μ0 ) − F (p3 |μ0 ) f (p3 |μ0 ) f (p4 |μ0 ) ln

(14) (15)

In order to make the function monotonicity, we know that p1 ≤ x1 , x2 ≤ p2 ≤ x3 , x2 ≤ p3 ≤ x3 , p4 ≥ x4 , where (x1 , y1), (x2 , y2), (x3 , y3 ) and (x4 , y4) are four extreme points of contaminated normal distribution’s LLR from left to right. Assume that z(x) = ln

f (x|μ1 ) , f (x|μ0 )

W1 = {(p1 , p2 ) : z(p1 ) = z(p2 )},

W2 = {(p3 , p4 ) : z(p3 ) = z(p4 )}, (16)

then the codomains are z(W1 ) = [min(y1 , y2), max(y1, y2 )], z(W2 ) = [min(y3 , y4), max(y3, y4 )].

(17) (18)

Proposition 1: z(W1 ) and z(W2 ) are disjoint (cf., Wu et al. (2017)). Proposition 2: There exists (p1 , p2 ) ∈ W1 , (p3 , p4 ) ∈ W2 , which satisfy Eq.(14) and Eq.(15) (cf., Wu et al. (2017)). Therefore, all contaminated normal distributions’ LLR can be transformed into RLLR2 based on the horizontal segment truncation. We can construct the RLCUSUM2 control chart based on the above RLLR2 S0# = 0,

# Sn# = max{0, Sn−1 + l# (xn )}.

(19)

However, RLLR2 did not keep rising trend, which may influence the performance of control charts. Insert Figure 5 about here.

3.4

Oblique line truncated RLCUSUM3 control chart

Horizontal segment truncation simply finds a fixed value in the non-monotone interval which cannot reflect the increasing trend of LLR. In order to overcome this disadvantage, Wu and Yu (2017) proposed the idea of oblique line truncation. They didn’t give an optimal design considering its complexity. In this paper, the improvements and optimal design of RLCUSUM control chart truncated by oblique line are given. We design the robust likelihood ratio (RLLR3 ) ⎧ ⎪ ⎨k(x − q1 ) + l(q1 ), i1 ≤ x ≤ i2 ,  (20) l (x) = k(x − q2 ) + l(q2 ), i3 ≤ x ≤ i4 , ⎪ ⎩ f (x|μ1 ) ln f (x|μ0 ) , Others. where x1 ≤ q1 ≤ x2 and x3 ≤ q2 ≤ x4 , k > 0, i1 ≤ x1 , x2 ≤ i2 ≤ x3 , x3 ≤ i3 ≤ x4 , i4 ≥ x4 and i1 , i2 , i3 , i4 are intersection points of diagonal segment and log likelihood ratio. The optimal 9

design will given in the next section. LLR is truncated by oblique line retains the tail information. At the same time, the original non-monotone intervals turn out to be strictly increasing. We can construct the RLCUSUM3 control chart based on the above RLLR3 , in which the statistics are S0 = 0,

4

 Sn = max{0, Sn−1 + l (xn )}.

(21)

The optimal design and enhancement of RLCUSUM3 control chart

In this section, the impacts of different parameters, such as the slope k and the point q1 , on the average run length are discussed to find the optimal design. Also, comparison among different control charts are also given. The enhancement to RLCUSUM control charts is considered, that is a fast initial response feature, to make the RLCUSUM control schemes more sensitive to start-up cases.

4.1

The optimal design of RLCUSUM3 control chart

In this sub-section, the optimal choice of parameters k, q1 and q2 in Eq. (20) is discussed. We use the ARL as the criterion to compare the performance of charts in different cases to decided what optimal values of parameters are. The control limits are given in subsection 4.1.1. To construct the oblique line, we should make choice of three parameters k, q1 and q2 . Considering the shape of log-likelihood ratio, we study the slope k ∈ [0.01, 0.1]. We consider the most representative computers in [x1 , x2 ]. The interval is equally divided into four parts 2 2 , d = x1 +3x , and the five sections are chosen, that are the a = x1 , b = 3x14+x2 , c = x1 +x 2 4 e = x2 . Because of the central symmetry, we consider the same points in [x3 , x4 ] and ⎧ ⎪ x4 , q1 = a, ⎪ ⎪ ⎪ x3 +3x4 ⎪ ⎪ ⎨ 4 , q1 = b, 4 (22) q2 = x3 +x , q1 = c, 2 ⎪ ⎪ 3x +x 3 4 ⎪ ⎪ ⎪ 4 , q1 = d, ⎪ ⎩x , q1 = e. 3 4.1.1

The control limits h under different μ1 given ARL0

Given μ0 = 0, σ02 = 1, and σ 2 = 9, the control limits h of RLCUSUM3 control charts under different μ1 and  are calculated with the method of Markov chain proposed by Brook and Evans (1972). The results are showed as follows. When i1 ≤ −100 or i4 ≥ 100, we assume that there is no such oblique line to construct the RLCUSUM3 control charts. Table 1 shows the control limits when μ1 = 1, 1.5, 2, · · · , 3.5 and  = 0.1 given the in-control average run length as 200, 370, 500, 1000. More extensive tables covering  = 0.05, 0.25 and 0.5 are available from the authors. 10

Insert Table 1 about here. 4.1.2

The optimal value of parameter k

In this subsection, we compare the the RLCUSUM3 control charts’ performances in detecting the various shifts (μ1 = 1, 2, 3, 3.5) when  = 0.05, 0.1, 0.25, 0.5 with the method of out-ofcontrol average run length (ARL1 ) given ARL0 = 1000. On one hand, even though the expectations of the null and alternative hypothesis are estimated, the mean of real process μ is unknown in the Phase II. If μ = μ0 , the process is in-control. If μ = μ1 , the designed control chart is optimal. On the other hand, since CUSUM control chart is mainly used in detecting the small shifts, there should not be a big difference among the process’s mean μ, μ0 and μ1 . Thus the cases of μ ∈ [0, 4] are considered in this section. Here we just show one case that  = 0.25 in the paper. More extensive figures for  = 0.05, 0.1 and 0.5 are available from the authors. In order to make the obvious difference we just have the part enlarged and results are showed in Figure 6-Figure 10. The contribution of slope k is very important for detecting various shifts. On one hand, in the case of small and medium shifts, larger k in RLCUSUM3 control charts would make a better performance, while there is quite the opposite to the case of large shifts that almost does not occur. On the other hand, the contribution of k has a greater influence which means that ΔARL1 := ARL1,ki − ARL1,kj , i, j = 1, 2, · · · , 10, is much bigger in detecting the small shifts under the same μ1 ,  and μ while the differences among the various slopes are much smaller under large shift cases. Due to the above mentioned, k = 0.06 or 0.07 is reasonable and in this paper, we choose k = 0.07. Insert Figure 6 about here. Insert Figure 7 about here. Insert Figure 8 about here. Insert Figure 9 about here. Insert Figure 10 about here. 4.1.3

The optimal value of parameter q1

From above sections, we conclude that k = 0.07 is a optimal value. In this subsection, we still use ARL1 given ARL0 = 1000 to select appropriate value of parameter q1 in detecting various shifts (μ1 = 1, 2, 3, 3.5) when  = 0.05, 0.1, 0.25, 0.5 and the mean of process μ ∈ [0, 4]. In order to make the obvious difference we just have the part enlarged, and results are showed in Figure 11-Figure 14. The contribution of point q1 is very important for detecting various shifts. 1. When μ1 is small, ARL1 in the case of q1 = a will decrease quickly with the increase of process’s mean μ while it is much larger when μ is small. Thus q1 = a is not a robust point. 2. For other four points, the closer q1 is to x2 , the better RLCUSUM3 control charts perform in the case of small and medium shifts while there would be a contrary situation when many large shifts happened, which is a fairly rare occurrence. Insert Figure 11 about here. 11

Insert Figure 12 about here. Insert Figure 13 about here. Insert Figure 14 about here. 3. Besides, the differences among the different q1 s are much smaller under large shift cases compared with small and medium shifts. 4. Furthermore, from Table 1, q1 = d is robust when we choose the optimal control limit h which is a benefit to the application in reality. On conclusion, q1 = d is a better choice.

4.2

Comparison with different truncated RLCUSUM control charts

In this section, we compare CUSUM, GCUSUM, and enhancements of RLCUSUM control charts’ performances when μ0 = 0, σ0 = 1, σ = 3σ0 = 3 in the case of μ1 = 1, 2, 3, 3.5 and  = 0.05, 0.1, 0.25, 0.5. Similarly, we use ARL1 given ARL0 = 1000 as the criterion. The results are shown in Figure 15 and Table 2. And we obtain that 1. It is unreasonable that the ARL1 of CUSUM and GCUSUM control charts are not monotonic decreasing with the increase of process’s mean μ. Robustification is desirable so that when the CUSUM value plots outside its control limits, it is likely to indicate a distributional shift rather than an isolated special cause (cf. Hawkins (1993)). And other three types of RLCUSUM control charts can overcome it. 2. Since the value of LLR tends to be limitless and the tail information is lost in RLCUSUM1 control chart, it performs much worse in detecting the large shifts compared with other RLCUSUM control charts. 3. When  and μ1 are small, RLCUSUM3 control chart is more sensitive in monitoring shifts in the mean compared to RLCUSUM2 control charts. Even though if its performance is little worse than RLCUSUM2 control chart when  and μ1 are large, there is a more rapid decrease in ARL1 of RLCUSUM3 control charts compared with RLCUSUM2 control charts. In general, RLCUSUM3 control charts have a better effect on monitoring for its relatively superior performances in various situations. Insert Table 2 about here. Insert Figure 15 about here.

4.3

RLCUSUM control chart with fast initial response feature

However, truncation methods cannot essentially improve the sensitivity of the CUSUM control charts in monitoring large shifts. Lucas and Crosier (1982a) showed that a FIR feature is useful for CUSUM control schemes because processes are more likely to be away from the target value when a control scheme is initiated due to startup problems or because of ineffective control action after the previous out-of-control signal (cf., Lucas and Saccucci (1990)). 12

With a moderate head-start, better properties are obtained in that the out-of-control average run length has a large percentage decrease while in-control average run length has a small percentage decrease. It means that an FIR CUSUM can be designed with an equivalent ARL0 and a smaller ARL1 than a standard CUSUM by increasing h slightly to compensate for the small decrease in ARL caused by the head start. If the process is out of control at first, non-zero initial head start value of CUSUM statistics, which is equivalent to lower control limits, can raise the probability of alarm. If initially the production process is in control, CUSUM statistics will be back down to zero fast. In this section, an optimal value of fast initial response is given, considering both the in-control and out-of-control cases. A FIR feature for all RLCUSUM control charts can be obtained and this paper just shows the case of RLCUSUM3 control chart. For the RLCUSUM3 control chart, we give it a head start γ as S0 = γ,

 Sn = max{0, Sn−1 + l (xn )},

(23)

which is called FRLCUSUM3 . Thus, it is important to determine γ under FRLCUSUM3 control chart, which is affected by the control limit h. Insert Table 3 about here. Similar with Lucas and Crosier (1982a), we present ARLs tables showing the run-length distribution for CUSUM control schemes with the FIR feature based on contaminated normal distribution. Table 3 shows the calculated ARLs of upward FRLCUSUM3 schemes with h values of 4 and 5, and head-start values of 0, h/4, h/2, 3h/4, and h. The selection values of the parameters are μ0 = 0, μ1 = 1, σ02 = 1, σ 2 = 9,  = 0.1. According to the criterion aforementioned, we recommend a head-start value of h/2, which is the same as shown in Lucas and Crosier (1982a). We get the similar results with  = 0.05, 0.25, 0.5 are showed in Table 3. More extensive tables covering different  are available from the authors. Insert Table 4 about here. In Table 4 we give the ARLs for upward schemes with μ0 = 0, μ1 = 1, σ02 = 1, σ 2 = 9. Respectively, the head start is set as our recommended h/2. More extensive tables covering head-start values and downward CUSUM control charts are available from the authors. In Figure 16 we present the comparison results about CUSUM, other three types of RLCUSUM, FRLCUSUM3 and GCUSUM control charts. More extensive figures covering  = 0.05, 0.25, 0.5 are available from the authors. The parameter value h of different control charts are shown in Table 5. The FRLCUSUM3 control charts have the head start value with γ = h/2. It can be seen that the ARL1 curve of both CUSUM and GCUSUM control charts are not monotonic decreasing with the increase of true mean shift, especially in Figure 16(a) and that can be overcome by other three types of RLCUSUM and FRLCUSUM3 control charts. Among the four RLCUSUM control charts, that have the monotonic decreasing ARL1 curves, we can conclude that FRLCUSUM3 control chart performs well for startup problems, which means that FIR feature is a good design when there is not clear about whether the process is initially out of control. Insert Table 5 about here. Insert Figure 16 about here.

13

5

An example

In this section, a real example is illustrated to show the implementation of RLCUSUM control scheme. We use a data set drawn from an actual production process, where a margarine company is packing 250-gram containers for retail sales using an automatic filling engine. It is a set of measurements of a packing container, which appeared in Yang et al. (2012) and Wadsworth et al. (1986). The data consist of 20 subsamples of size 5, i.e., 100 observations. The mean of the process is 250 g, and the standard deviation, σ = 1.393, is estimated from the samples. The real data set is standardized here. Let all the data minus its mean 250 to get the standardized data set, which is showed in Figure 17(a). The standardized data have mean 0.76, and it also can be seen in the Figure 17(a) that the data are out-of-control at the start, which means that the data have an overall upward shift in the mean. Insert Figure 17 about here. We use a contaminated normal distribution to fit the data. Figure 17(b) shows the comparison results between the real density and the fitting density, where the fitting density is f (x) = (1 − )φ(x; μ, σ02 ) + φ(x; μ, σ 2 ) (24) with μ = 0, σ02 = 1, σ 2 = 9,  = 0.1 and φ(x; μ, σ 2) is the normal probability density function with mean μ and variance σ 2 . Figure 17(b) shows the density comparison results under the null hypothesis μ = 0 and the alternative hypothesis μ = 1. It is concluded that the fitted density curve is closer to the sample density curve, which means that the data have an obvious upward shift. The density comparison results support the later analysis of those adjusted CUSUM control charts. Then we apply the aforementioned control charts to the data set. The values of parameter h of different control charts are the same as the first row of Table 5 with μ1 = 1. The headstart values of FRLCUSUM3 are γ = h/2. Figure 18(a)-(f) shows the comparison results. We can see that CUSUM, RLCUSUM1 , RLCUSUM2 and RLCUSUM3 all give a signal at the 8th observation. The data set have mean 0.76, which is a small shift. However, comparing with CUSUM, the common advantage of RLCUSUM1 , RLCUSUM2 and RLCUSUM3 is for large shifts, so those charts show same signals here. Though the GCUSUM control chart signals faster than above-mentioned control charts, it doesn’t perform so robustly. It doesn’t give the signals at the 5th, 6th and 7th observations, which is obvious in contrast to the fact compared with Figure 17(a). Only the FRLCUSUM3 signals at the 3rd observation. Considering the process is out-of-control at start, the FRLCUSUM3 control chart performs better than others here. Insert Figure 18 about here.

6

Conclusions

This paper evaluates the properties of RLCUSUM control schemes used to monitor the mean of contaminated normally distributed processes. An optimal design procedure for RLCUSUM3 control schemes is given to construct the RLLR3 appropriate for kinds of shifts. We give the reference values that slope k = 0.07 and point q1 = d for two reasons: first, 14

under this selection, control limits h are basically stable in different kinds of shifts that helps to RLCUSUM3 control chart’s application. Second, we should ensure the smaller of ARL1 when ARL0 is fixed in various shifts. An extensive comparison with the conventional CUSUM and generalized likelihood ratio CUSUM in terms of average run length, shows that RLCUSUM3 control chart are better in following items: 1. With the increase of process’s mean μ, it gives an alarm more rapidly which is almost impossible to achieve in CUSUM and GCUSUM control charts. 2. It can surely be structured with retaining the tail information compared with RLCUSUM1 control charts. 3. When μ1 and  are small, it is more sensitive in detecting large shifts compared with RLCUSUM2 control charts. 4. Most of the time, it is more robust and sensitive compared with other control charts. RLCUSUM control chart detects small shifts at the cost of little sensitive for constructing the RLLR. The enhancement to RLCUSUM is considered, that is a fast initial response feature that makes the RLCUSUM control schemes more sensitive to start-up cases. We find that S0 = h2 is a best head-start value, since it helps ARL1 to decline much when the process is out of control while FIR feature increases the number of very short run lengths when the process in control. The expected value and a final application based on monitoring packing 250-gram containers both have proven that FRLCUSUM3 control charts have a more sensitive effect of detecting small shifts. Insert Table 6 about here.

A

Appendix A: Notations

Insert Table 7 about here.

References Brook, D., & Evans, D. (1972). An approach to the probability distribution of CUSUM run length. Biometrika, 59(3):539–549. Chatterjee, S., & Qiu, P. (2009). Distribution-free cumulative sum control charts using bootstrap-based control limits. The Annals of Applied Statistics, 3(1):349–369. Hackl, P., & Maderbacher, M. (1999). On the robustness of the rank-based CUSUM chart against autocorrelation. Forschungsberichte. Han, D., & Tsung, F. (2004). A generalized EWMA control chart and its comparison with the optimal EWMA, CUSUM and GLR schemes. The Annals of Statistics, 32(1):316–339. Hawkins, D. M. (1993). Robustification of cumulative sum charts by winsorization. Journal of Quality Technology, 25(4):248–261. 15

Li, S., Tang, L. C., & Ng, S. H. (2010). Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology, 42(2):209–226. Liu, G., Pu, X., Wang, L., & Xiang, D. (2015). CUSUM chart for detecting range shifts when monotonicity of likelihood ratio is invalid. Journal of Applied Statistics, 42(8):1635–1644. Lucas, J. M. (1982). Combined shewhart-CUSUM quality control schemes. Journal of quality technology, 14(2):51–59. Lucas, J. M., & Crosier, R. B. (1982a). Fast initial response for CUSUM quality-control schemes: Give your cusum a head start. Technometrics, 24(3):199–205. Lucas, J. M., & Crosier, R. B. (1982b). Robust CUSUM: A robust study for CUSUM quality control schemes. Communication in Statistics Theory and Methods, 11(23):2669–2687. Lucas, J. M., & Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: Properties and enhancements. Technometrics, 32(1):199–205. MacEachern, S. N., Rao, Y., & Wu, C. (2007). A robust-likelihood cumulative sum chart. Journal of the American Statistical Association, 102(480):1440–1447. Nazir, H. Z., Riaz, M., Does, R. J. M. M., & Abbas, N. (2013). Robust CUSUM control charting. Quality Engineering, 25(3):211–224. Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1-2):223–233. Reynolds, J. M. R., & Stoumbos, Z. G. (2010). Robust CUSUM charts for monitoring the process mean and variance. Quality and Reliability Engineering International, 26(5):453– 473. Siegmund, D., & Venkatraman, E. S. (1995). Using the generailized likelihood ratio statistic for sequential detection of a change-point. The Annals of Statistics, 23(1):255–271. Wadsworth, H. M., Stephens, K. S., & Godfrey, A. B. (1986). Modern methods for quality control and improvement. Wiley, (Chapter 7). Westgard, J. O., Groth, T., Aronsson, T., & de Verdier, C. H. (1977). Combined shewhartCUSUM control chart for improved quality control in clinical chemistry. Clinical Chemistry, 23(10):1881-1887. Wu, C., Wei, Y., & Yu, M. (2017). Improved robust-likelihood cumulative sum chart for the contaminated normal distributions (in chinese). Science China Mathematics, 47(7):853868. Wu, C., & Yu, M. (2017). The robust-likelihood cumulative sum control chart cut off by diagonal when observations followed contaminated normal distribution (in chinese) . Journal of System Science and Mathematical Science, 37(4):1138-1155.

16

Wu, C., Zhao, Y., & Wang, Z. (2002). The median absolute deviations and theri applications to shewhart control charts. Communications in Statistics - Simulation and Computation, 31(3):425–442. Wu, Z., Yang, M., Jiang, W., & Khoo, M. B. C. (2008). Optimization designs of the combined shewhart-cusum control charts. Computational Statistics and Data Analysis, 53(2):496–506. Yang, L., Pai, S., & Wang, Y. R. (2010). A novel CUSUM median control chart. Lecture Notes in Engineering and Computer Science, 3:1707-1710. Yang, L., Wang, Y., & Pai, S. (2012). Statistical and economic analyses of an EWMA-based synthesised control scheme for monitoring processes with outliers. International Journal of Systems Science, 43(2):285–295. Yang, S. F., & Cheng, S. W. (2011). A new nonparametric CUSUM mean chart. Quality and Reliability Engineering International, 27(7):867–875.

17

Figure captions: Figure 1. LLR under different , where μ0 = 0, μ1 = 1, σ02 = 1, σ 2 = 1. Figure 2. LLR under different μ1 , where μ0 = 0, σ02 = 1, σ 2 = 9,  = 0.05. Figure 3. g(x) under different μ1 and , where μ0 = 0, σ02 = 1, σ 2 = 4. Figure 4. g(x) under different μ1 and , where μ0 = 0, σ02 = 1, σ 2 = 9. Figure 5. The log-likelihood ratio and RLLR for contaminated normal distribution when μ0 = 0, μ1 = 1, σ02 = 1, σ 2 = 9,  = 0.05. Figure 6. The ARL1 when  = 0.25, q1 = a given ARL0 = 1000. Figure 7. The ARL1 when  = 0.25, q1 = b given ARL0 = 1000. Figure 8. The ARL1 when  = 0.25, q1 = c given ARL0 = 1000. Figure 9. The ARL1 when  = 0.25, q1 = d given ARL0 = 1000. Figure 10. The ARL1 when  = 0.25, q1 = e given ARL0 = 1000. Figure 11. The ARL1 when  = 0.05, k = 0.07 given ARL0 = 1000. Figure 12. The ARL1 when  = 0.1, k = 0.07 given ARL0 = 1000. Figure 13. The ARL1 when  = 0.25, k = 0.07 given ARL0 = 1000. Figure 14. The ARL1 when  = 0.5, k = 0.07 given ARL0 = 1000. Figure 15. The ARL1 of five control charts under different means of process μ given ARL0 = 1000. Figure 16. Comparison results of different CUSUM control charts. Figure 17. Density comparison plots of real data. Figure 18. Comparison of different CUSUM control charts.

18

19

μ1

=a =b =c =d =e

=a =b =c =d =e

=a =b =c =d =e

=a =b =c =d =e

=a =b =c =d =e

=a =b =c =d =e

=a =b =c =d =e

=a =b =c =d =e

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

q1 q1 q1 q1 q1

k = 0.02

k = 0.03

k = 0.04

k = 0.05

k = 0.06

k = 0.07

k = 0.08

k = 0.09

k = 0.1

# # # # #

# # # # #

# # # # #

# # # # #

# # # # #

0.63 0.98 1.91 # #

0.80 1.11 1.97 2.74 2.94

0.97 1.24 2.02 2.74 2.92

1.13 1.36 2.07 2.74 2.91

1.28 1.46 2.11 2.74 2.90

200

# # # # #

# # # # #

# # # # #

# # # # #

# # # # #

0.77 1.19 2.30 # #

0.97 1.34 2.37 3.29 3.54

1.17 1.49 2.43 3.29 3.52

1.36 1.63 2.49 3.29 3.50

# # # # #

# # # # #

# # # # #

# # # # #

# # # # #

0.83 1.29 2.50 # #

1.05 1.46 2.57 3.57 3.84

1.27 1.62 2.64 3.57 3.82

1.48 1.77 2.70 3.57 3.80

1.67 1.91 2.76 3.57 3.79

500

0.5

1.54 1.76 2.54 3.29 3.49

370

# # # # #

# # # # #

# # # # #

# # # # #

# # # # #

1.00 1.54 2.97 # #

1.25 1.73 3.05 4.24 4.56

1.51 1.92 3.13 4.24 4.54

1.76 2.11 3.21 4.24 4.52

2.00 2.27 3.27 4.24 4.50

1000

0.87 1.35 2.53 3.49 3.91

0.98 1.43 2.56 3.54 3.83

1.09 1.51 2.60 3.54 3.82

1.20 1.59 2.63 3.54 3.81

1.30 1.67 2.66 3.54 3.80

1.41 1.76 2.69 3.54 3.79

1.52 1.83 2.72 3.54 3.78

1.63 1.90 2.74 3.54 3.77

1.72 1.97 2.77 3.54 3.76

1.81 2.03 2.79 3.54 3.75

200

1.03 1.58 2.96 4.10 4.63

1.15 1.67 3.00 4.15 4.49

1.27 1.77 3.04 4.15 4.48

1.40 1.86 3.08 4.15 4.47

1.53 1.95 3.12 4.15 4.45

1.66 2.04 3.15 4.15 4.44

1.77 2.12 3.19 4.15 4.43

1.89 2.21 3.22 4.14 4.42

2.00 2.28 3.25 4.14 4.40

2.10 2.36 3.28 4.14 4.39

370

1

1.11 1.69 3.17 4.40 4.97

1.23 1.79 3.21 4.45 4.82

1.37 1.89 3.25 4.45 4.81

1.50 1.99 3.30 4.44 4.79

1.64 2.09 3.34 4.44 4.78

1.77 2.19 3.37 4.44 4.76

1.90 2.28 3.41 4.44 4.75

2.03 2.37 3.45 4.44 4.74

2.15 2.46 3.48 4.44 4.72

2.26 2.54 3.52 4.43 4.71

500

1.29 1.96 3.66 5.11 5.81

1.43 2.08 3.71 5.14 5.58

1.58 2.19 3.76 5.14 5.56

1.74 2.31 3.81 5.14 5.55

1.90 2.42 3.86 5.14 5.53

2.05 2.53 3.90 5.14 5.51

2.20 2.63 3.94 5.14 5.50

2.34 2.74 3.98 5.14 5.48

2.48 2.83 4.02 5.13 5.46

2.60 2.93 4.06 5.13 5.45

1000

1.41 1.87 2.93 3.91 4.21

1.49 1.93 2.94 3.92 4.20

1.57 2.00 2.96 3.92 4.19

1.65 2.06 2.98 3.92 4.18

1.74 2.11 2.99 3.92 4.17

1.83 2.15 3.01 3.92 4.16

1.92 2.20 3.03 3.91 4.15

1.98 2.25 3.04 3.91 4.15

2.04 2.29 3.06 3.91 4.14

2.10 2.33 3.08 3.91 4.13

200

1.64 2.14 3.43 4.47 4.87

1.73 2.21 3.45 4.47 4.86

1.81 2.27 3.48 4.46 4.85

1.90 2.33 3.50 4.46 4.84

1.98 2.38 3.52 4.46 4.83

2.05 2.44 3.54 4.45 4.81

2.13 2.50 3.56 4.45 4.80

2.20 2.56 3.58 4.45 4.79

2.27 2.62 3.61 4.45 4.77

1.75 2.30 3.65 4.78 5.18

1.85 2.36 3.68 4.78 5.17

1.95 2.42 3.71 4.78 5.15

2.04 2.48 3.74 4.78 5.14

2.13 2.54 3.77 4.77 5.12

2.21 2.60 3.79 4.77 5.11

2.29 2.66 3.82 4.77 5.10

2.37 2.71 3.84 4.77 5.08

2.45 2.77 3.86 4.77 5.07

2.53 2.82 3.88 4.77 5.06

500

1.5

2.34 2.69 3.62 4.44 4.76

370

2.01 2.62 4.16 5.51 5.96

2.11 2.70 4.19 5.50 5.95

2.22 2.78 4.22 5.50 5.94

2.32 2.87 4.24 5.50 5.92

2.43 2.95 4.27 5.50 5.90

2.54 3.03 4.29 5.50 5.89

2.64 3.09 4.32 5.49 5.88

2.76 3.16 4.34 5.49 5.86

2.86 3.23 4.37 5.49 5.85

2.95 3.29 4.39 5.49 5.84

1000

1.71 2.13 3.29 3.99 4.26

1.76 2.17 3.31 3.99 4.25

1.81 2.21 3.32 3.99 4.25

1.86 2.25 3.33 3.99 4.24

1.91 2.29 3.34 3.99 4.23

1.96 2.33 3.36 3.98 4.22

2.00 2.37 3.37 3.98 4.21

2.05 2.41 3.38 3.98 4.21

2.10 2.46 3.39 3.98 4.20

2.15 2.51 3.40 3.98 4.19

200

2.03 2.47 3.62 4.71 5.03

2.09 2.50 3.64 4.71 5.02

2.15 2.54 3.65 4.70 5.01

2.20 2.57 3.67 4.70 5.00

2.26 2.61 3.68 4.70 4.99

2.31 2.64 3.70 4.70 4.98

2.36 2.68 3.71 4.70 4.97

2.41 2.72 3.73 4.70 4.97

2.47 2.75 3.75 4.69 4.96

2.52 2.79 3.76 4.69 4.95

370

2

2.16 2.67 3.83 5.03 5.40

2.24 2.71 3.84 5.04 5.39

2.33 2.75 3.85 5.04 5.38

2.40 2.80 3.86 5.04 5.37

2.46 2.84 3.87 5.04 5.36

2.52 2.88 3.88 5.04 5.35

2.58 2.92 3.88 5.03 5.34

2.63 2.96 3.89 5.03 5.33

2.69 3.00 3.90 5.03 5.32

2.74 3.04 3.92 5.03 5.31

500

2.43 3.03 4.44 5.59 6.13

2.51 3.09 4.46 5.58 6.11

2.58 3.16 4.48 5.57 6.10

2.66 3.23 4.50 5.56 6.08

2.74 3.30 4.52 5.55 6.06

2.82 3.37 4.53 5.55 6.04

2.91 3.44 4.55 5.54 6.02

3.00 3.49 4.57 5.54 6.00

3.09 3.53 4.59 5.54 5.98

3.19 3.58 4.60 5.53 5.96

1000

2.05 2.48 3.18 3.77 4.03

2.12 2.51 3.19 3.77 4.02

2.19 2.53 3.20 3.77 4.01

2.26 2.55 3.21 3.77 4.00

2.32 2.58 3.22 3.77 3.99

2.35 2.60 3.22 3.76 3.98

2.38 2.62 3.23 3.76 3.97

2.41 2.64 3.24 3.76 3.97

2.44 2.66 3.25 3.76 3.96

2.47 2.68 3.26 3.76 3.95

200

2.24 2.77 3.92 4.61 4.89

2.29 2.82 3.93 4.61 4.89

2.35 2.86 3.94 4.61 4.88

2.41 2.91 3.95 4.60 4.87

2.46 2.97 3.96 4.60 4.86

2.52 3.02 3.87 4.60 4.85

2.58 3.07 3.98 4.60 4.84

2.65 3.13 3.99 4.60 4.84

2.71 3.19 4.00 4.60 4.83

2.35 2.89 4.24 5.01 5.31

2.40 2.92 4.27 5.01 5.30

2.45 2.96 4.28 5.00 5.30

2.50 3.00 4.29 5.00 5.29

2.55 3.04 4.31 5.00 5.28

2.60 3.09 4.32 5.00 5.27

2.64 3.13 4.33 5.00 5.26

2.69 3.17 4.34 4.99 5.25

2.75 3.22 4.35 4.99 5.24

2.80 3.27 4.36 4.99 5.23

500

2.5

2.78 3.25 4.01 4.59 4.82

370

NOTE: # means that there is no such oblique line to construct the RLCUSUM3 control charts.

=a =b =c =d =e

q1 q1 q1 q1 q1

k = 0.01

1

=a =b =c =d =e

q1 q1 q1 q1 q1

ARL0

2.72 3.23 4.58 5.89 6.25

2.77 3.25 4.59 5.89 6.24

2.82 3.28 4.61 5.89 6.23

2.87 3.31 4.62 5.88 6.22

2.92 3.34 4.63 5.88 6.21

2.97 3.37 4.65 5.88 6.20

3.02 3.40 4.66 5.88 6.19

3.07 3.43 4.68 5.88 6.18

3.12 3.46 4.69 5.87 6.17

3.16 3.50 4.71 5.87 6.16

1000

1.98 2.24 3.00 3.71 4.05

2.00 2.24 2.98 3.69 4.03

2.02 2.24 2.97 3.67 4.01

2.04 2.25 2.96 3.66 3.98

2.06 2.26 2.94 3.64 3.96

2.08 2.28 2.93 3.62 3.93

2.10 2.29 2.91 3.59 3.91

2.12 2.31 2.89 3.57 3.88

2.14 2.32 2.87 3.55 3.85

2.16 2.34 2.84 3.52 3.82

200

2.62 2.94 3.65 4.23 4.48

2.65 2.96 3.66 4.23 4.46

2.68 2.98 3.66 4.22 4.45

2.71 3.00 3.67 4.22 4.44

2.74 3.02 3.68 4.22 4.43

2.77 3.04 3.69 4.22 4.43

2.80 3.06 3.70 4.22 4.42

2.83 3.08 3.70 4.22 4.41

2.86 3.10 3.71 4.22 4.40

2.88 3.12 3.72 4.21 4.40

370

3

2.74 3.31 4.07 4.69 4.94

2.81 3.33 4.08 4.69 4.94

2.89 3.35 4.09 4.69 4.93

2.96 3.37 4.10 4.69 4.92

3.04 3.40 4.11 4.69 4.91

3.11 3.42 4.12 4.68 4.90

3.15 3.44 4.13 4.68 4.90

3.18 3.46 4.13 4.68 4.89

3.21 3.48 4.14 4.68 4.88

3.24 3.50 4.15 4.68 4.87

500

Table 1: Control limits h under different μ1 when μ0 = 0, σ02 = 1, σ 2 = 9,  = 0.1

2.97 3.60 5.02 5.74 6.03

3.02 3.65 5.03 5.74 6.03

3.07 3.69 5.04 5.74 6.02

3.13 3.74 5.05 5.74 6.01

3.18 3.79 5.06 5.74 6.00

3.24 3.84 5.07 5.74 5.99

3.30 3.89 5.08 5.73 5.98

3.36 3.94 5.09 5.73 5.97

3.42 3.99 5.10 5.73 5.96

3.48 4.04 5.12 5.73 5.95

1000

2.04 2.41 3.21 3.91 4.22

2.06 2.41 3.21 3.90 4.20

2.07 2.42 3.21 3.89 4.19

2.08 2.42 3.21 3.88 4.18

2.09 2.43 3.20 3.88 4.16

2.10 2.43 3.20 3.87 4.15

2.11 2.43 3.20 3.86 4.13

2.12 2.44 3.20 3.85 4.12

2.13 2.44 3.20 3.84 4.10

2.14 2.44 3.19 3.84 4.09

200

2.37 2.63 3.40 4.08 4.38

2.38 2.62 3.39 4.06 4.35

2.40 2.62 3.37 4.03 4.32

2.42 2.62 3.36 4.01 4.29

2.44 2.63 3.34 3.99 4.27

2.45 2.64 3.32 3.96 4.24

2.47 2.66 3.30 3.94 4.21

2.49 2.67 3.28 3.91 4.18

2.50 2.68 3.25 3.89 4.14

2.70 2.97 3.62 4.23 4.51

2.73 2.98 3.61 4.21 4.48

2.75 3.00 3.60 4.18 4.44

2.77 3.02 3.60 4.16 4.41

2.79 3.03 3.61 4.13 4.38

2.81 3.05 3.62 4.11 4.34

2.84 3.06 3.63 4.10 4.31

2.86 3.08 3.64 4.09 4.27

2.88 3.10 3.64 4.09 4.25

2.90 3.11 3.65 4.09 4.24

500

3.5

2.52 2.69 3.22 3.86 4.11

370

3.44 3.96 4.72 5.31 5.54

3.52 3.98 4.73 5.31 5.53

3.59 4.01 4.74 5.31 5.52

3.67 4.03 4.75 5.31 5.51

3.72 4.05 4.76 5.31 5.50

3.75 4.07 4.77 5.31 5.50

3.78 4.09 4.78 5.30 5.49

3.82 4.11 4.79 5.30 5.48

3.85 4.13 4.80 5.30 5.47

3.88 4.15 4.80 5.30 5.47

1000

20

46.42 12.97 7.52 5.70 5.04 4.99 5.39 6.15 7.11 7.91 8.27 8.15 7.78 7.32 6.90 6.49 6.11 5.86 5.53 5.19

57.03 16.60 9.82 7.62 6.89 6.88 7.32 7.95 8.47 8.63 8.43 8.02 7.56 7.09 6.68 6.29 5.97 5.68 5.36 5.11

80.33 25.23 15.09 11.66 10.30 9.80 9.63 9.46 9.13 8.67 8.13 7.60 7.11 6.68 6.28 5.94 5.62 5.33 5.08 4.86

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5  = 0.1 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5  = 0.25 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5  = 0.5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

81.91 25.86 15.34 11.61 9.87 8.95 8.39 8.01 7.74 7.54 7.38 7.27 7.18 7.12 7.08 7.05 7.03 7.02 7.01 7.01

58.63 17.18 10.11 7.67 6.54 5.94 5.61 5.42 5.31 5.23 5.17 5.12 5.08 5.06 5.04 5.03 5.02 5.01 5.01 5.00

47.63 13.34 7.73 5.78 4.89 4.45 4.23 4.14 4.09 4.07 4.05 4.03 4.02 4.02 4.01 4.01 4.00 4.00 4.00 4.00

83.45 26.15 15.49 11.71 9.94 8.97 8.37 7.96 7.65 7.40 7.17 6.95 6.71 6.43 6.12 5.83 5.55 5.28 5.04 4.84

58.70 17.19 10.12 7.67 6.54 5.94 5.61 5.42 5.31 5.23 5.16 5.12 5.08 5.06 5.04 5.02 5.01 5.00 4.98 4.95

47.72 13.36 7.74 5.79 4.90 4.45 4.23 4.14 4.09 4.07 4.05 4.03 4.02 4.02 4.01 4.01 4.00 4.00 4.00 4.00

84.78 26.74 15.69 11.67 9.72 8.60 7.86 7.28 6.81 6.46 6.20 5.96 5.72 5.46 5.26 5.11 5.02 4.94 4.85 4.71

60.30 17.61 10.28 7.71 6.50 5.84 5.49 5.31 5.18 5.05 4.89 4.64 4.36 4.17 4.07 4.03 4.01 4.01 4.00 3.99

48.10 13.46 7.77 5.78 4.85 4.40 4.19 4.10 4.07 4.04 4.01 3.97 3.88 3.67 3.38 3.15 3.05 3.02 3.01 3.00

158.50 35.69 14.96 9.41 7.32 6.41 6.01 5.81 5.66 5.48 5.22 4.93 4.64 4.36 4.10 3.88 3.69 3.49 3.30 3.15

127.66 24.41 9.72 6.06 4.75 4.26 4.20 4.36 4.63 4.86 4.94 4.87 4.70 4.45 4.17 3.97 3.80 3.59 3.35 3.16

107.27 19.06 7.45 4.59 3.54 3.14 3.08 3.24 3.59 4.03 4.42 4.65 4.66 4.47 4.17 3.99 3.85 3.63 3.34 3.13

153.61 35.52 15.09 9.55 7.45 6.50 6.01 5.71 5.51 5.37 5.26 5.18 5.12 5.08 5.05 5.03 5.02 5.01 5.01 5.00

131.54 25.38 10.11 6.26 4.78 4.04 3.63 3.39 3.27 3.19 3.14 3.10 3.07 3.05 3.04 3.03 3.02 3.01 3.01 3.00

105.26 19.13 7.57 4.72 3.69 3.29 3.14 3.08 3.06 3.04 3.03 3.02 3.01 3.01 3.01 3.00 3.00 3.00 3.00 3.00

154.13 35.58 15.09 9.52 7.38 6.38 5.83 5.47 5.20 4.97 4.76 4.54 4.31 4.10 3.94 3.79 3.62 3.43 3.25 3.11

131.43 25.37 10.11 6.26 4.77 4.04 3.63 3.39 3.26 3.19 3.14 3.10 3.07 3.05 3.04 3.02 3.01 3.01 3.00 2.99

105.26 19.13 7.57 4.72 3.69 3.29 3.14 3.08 3.06 3.04 3.03 3.02 3.01 3.01 3.00 3.00 3.00 2.99 2.99 2.98

160.13 36.83 15.51 9.69 7.40 6.27 5.59 5.09 4.73 4.48 4.31 4.19 4.10 4.02 3.94 3.86 3.76 3.63 3.46 3.28

135.38 26.03 10.32 6.39 4.88 4.11 3.65 3.39 3.25 3.17 3.12 3.09 3.06 3.04 3.03 3.02 3.01 3.00 3.00 2.99

107.93 19.49 7.66 4.75 3.70 3.28 3.12 3.05 3.00 2.92 2.77 2.53 2.29 2.13 2.05 2.02 2.01 2.01 2.00 2.00

282.15 73.23 23.60 11.29 7.31 5.66 4.88 4.51 4.30 4.15 3.99 3.81 3.60 3.37 3.16 3.00 2.87 2.75 2.60 2.44

262.44 56.64 16.00 7.33 4.75 3.76 3.36 3.22 3.20 3.24 3.32 3.39 3.36 3.23 3.08 2.98 2.90 2.79 2.62 2.41

227.01 43.84 12.02 5.47 3.50 2.72 2.38 2.27 2.33 2.51 2.74 2.93 3.03 3.03 3.00 2.96 2.92 2.82 2.62 2.38

298.72 74.46 23.87 11.47 7.49 5.88 5.14 4.76 4.53 4.38 4.27 4.19 4.13 4.09 4.06 4.04 4.02 4.02 4.01 4.01

232.11 51.16 15.22 7.18 4.74 3.82 3.43 3.27 3.19 3.14 3.10 3.07 3.05 3.03 3.02 3.02 3.01 3.01 3.00 3.00

234.33 45.77 12.46 5.64 3.60 2.76 2.37 2.18 2.09 2.06 2.04 2.03 2.02 2.01 2.01 2.01 2.00 2.00 2.00 2.00

269.15 69.96 23.02 11.16 7.30 5.71 4.94 4.51 4.22 3.99 3.78 3.56 3.35 3.16 3.02 2.92 2.81 2.69 2.54 2.38

233.92 51.45 15.27 7.20 4.75 3.82 3.43 3.26 3.17 3.11 3.06 3.02 2.98 2.95 2.90 2.83 2.72 2.54 2.34 2.17

234.32 45.77 12.46 5.64 3.60 2.76 2.37 2.18 2.09 2.06 2.04 2.03 2.02 2.01 2.01 2.01 2.00 2.00 2.00 2.00

278.21 73.10 23.88 11.49 7.45 5.75 4.89 4.35 3.95 3.64 3.42 3.27 3.17 3.09 3.02 2.95 2.89 2.80 2.69 2.54

244.11 54.32 15.90 7.40 4.83 3.84 3.42 3.23 3.11 3.01 2.88 2.69 2.46 2.26 2.14 2.08 2.05 2.03 2.01 2.01

238.50 46.77 12.65 5.70 3.62 2.78 2.37 2.18 2.09 2.05 2.04 2.03 2.02 2.01 2.01 2.01 2.00 2.00 2.00 2.00

353.30 108.81 34.33 14.33 8.24 5.85 4.72 4.11 3.76 3.56 3.43 3.31 3.17 3.04 2.93 2.81 2.67 2.51 2.35 2.22

376.08 100.22 25.86 9.72 5.46 3.92 3.25 2.96 2.91 2.96 3.02 3.04 3.02 2.99 2.94 2.86 2.72 2.52 2.31 2.16

280.99 63.49 16.56 6.55 3.78 2.76 2.34 2.16 2.10 2.11 2.19 2.35 2.55 2.69 2.71 2.59 2.38 2.18 2.06 2.01

352.56 109.12 34.52 14.42 8.30 5.90 4.75 4.12 3.76 3.54 3.39 3.29 3.21 3.15 3.11 3.07 3.05 3.03 3.02 3.01

452.23 101.81 25.63 9.76 5.58 4.13 3.55 3.32 3.21 3.15 3.11 3.08 3.05 3.04 3.03 3.02 3.01 3.01 3.00 3.00

280.53 64.02 16.81 6.65 3.83 2.80 2.36 2.16 2.08 2.05 2.04 2.03 2.02 2.01 2.01 2.01 2.00 2.00 2.00 2.00

354.51 108.85 34.36 14.37 8.27 5.88 4.73 4.10 3.73 3.50 3.34 3.21 3.11 3.01 2.91 2.80 2.67 2.51 2.35 2.22

441.76 100.67 25.50 9.73 5.56 4.11 3.54 3.29 3.17 3.10 3.04 2.99 2.94 2.88 2.79 2.64 2.45 2.26 2.12 2.04

280.50 64.01 16.81 6.65 3.83 2.80 2.36 2.16 2.08 2.05 2.04 2.03 2.02 2.01 2.01 2.01 2.00 2.00 2.00 2.00

348.02 107.42 34.25 14.40 8.30 5.90 4.73 4.09 3.71 3.48 3.32 3.19 3.09 3.00 2.92 2.82 2.70 2.55 2.39 2.25

361.64 98.14 26.02 9.91 5.62 4.09 3.43 3.07 2.77 2.49 2.28 2.16 2.10 2.07 2.05 2.03 2.02 2.01 2.01 2.00

283.22 65.17 17.07 6.72 3.86 2.81 2.36 2.16 2.08 2.05 2.04 2.03 2.02 2.01 2.01 2.01 2.00 2.00 2.00 2.00

44.38 17.55 10.21 6.70 5.27 4.27 3.62 3.16 2.79 2.52 2.33 2.21 2.13 2.07 2.01 1.97 1.92 1.88 1.82 1.73

23.69 8.88 5.26 3.72 2.92 2.48 2.22 2.03 1.86 1.70 1.59 1.57 1.62 1.70 1.75 1.75 1.66 1.52 1.37 1.23

22.76 7.92 4.46 3.06 2.34 1.91 1.62 1.39 1.24 1.14 1.11 1.14 1.23 1.37 1.53 1.65 1.68 1.62 1.47 1.31

μ1 = 1 μ1 = 2 μ1 = 3 μ1 = 3.5 μ CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 WCUSUM

Table 2: ARL1 given ARL0 = 1000 of four types of control charts

Table 3: ARLs for FRLCUSUM3 control charts under contaminated normal distribution μ

0

h/4

h=4 h/2

3h/4

h

0

h/4

h=5 h/2

3h/4

h

0 0.25 0.5 0.75 1 1.5 2 2.5 3 4 5

318.34 85.71 31.72 16.24 10.44 6.16 4.59 3.83 3.42 3.11 3.04

313.90 82.34 29.13 14.21 8.80 5.03 3.78 3.29 3.07 2.70 2.22

298.21 73.66 24.05 10.97 6.54 3.64 2.70 2.31 2.15 2.05 2.02

248.97 55.17 16.04 6.77 3.90 2.11 1.52 1.26 1.13 1.04 1.02

161.16 30.75 8.00 3.34 2.05 1.33 1.13 1.06 1.03 1.01 1.01

868.95 159.01 45.91 21.15 13.09 7.58 5.64 4.76 4.34 4.09 4.02

862.03 154.07 42.28 18.39 10.91 6.05 4.43 3.69 3.33 3.09 3.04

830.88 139.40 34.66 13.90 7.93 4.30 3.11 2.56 2.28 2.08 2.03

708.23 104.79 22.51 8.24 4.54 2.41 1.70 1.37 1.19 1.05 1.02

430.15 52.40 9.63 3.47 2.07 1.33 1.13 1.06 1.03 1.01 1.01

Table 4: ARLs for FRLCUSUM3 control charts under contaminated normal distribution μ

parameters h 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

 0.05 0.05 0.05 0.05 0.05 0.10 0.10 0.10 0.10 0.10 0.25 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.50 0.50

γ 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5

0 9.29 34.4 105. 313. 887. 8.59 32.9 104. 298. 831. 7.15 30.2 98.2 275. 733. 9.51 36.4 109. 295. 752.

0.25 5.90 15.8 35.0 72.0 136. 5.69 15.9 36.7 73.7 139. 5.22 16.4 39.2 79.0 148. 6.88 21.2 48.8 98.4 183.

0.5 3.97 8.26 14.3 22.6 32.4 3.93 8.51 15.5 24.1 34.7 3.85 9.43 18.0 28.4 41.6 5.08 13.1 24.5 39.7 58.6

0.75 2.85 4.93 7.36 10.2 12.8 2.86 5.15 8.15 11.0 13.9 2.92 5.95 9.85 13.6 17.6 3.87 8.72 14.1 20.2 26.4

1 2.18 3.33 4.59 6.03 7.25 2.21 3.49 5.19 6.54 7.93 2.31 4.12 6.35 8.21 10.2 3.05 6.27 9.35 12.5 15.6

1.5 1.50 1.98 2.55 3.37 3.92 1.53 2.07 3.06 3.64 4.30 1.63 2.47 3.71 4.55 5.60 2.11 3.98 5.47 7.00 8.54

2 1.22 1.46 1.80 2.53 2.85 1.24 1.52 2.39 2.70 3.11 1.32 1.79 2.80 3.30 4.17 1.65 3.05 4.01 5.10 6.22

2.5 1.09 1.22 1.43 2.20 2.38 1.11 1.26 2.12 2.31 2.56 1.17 1.45 2.42 2.72 3.58 1.41 2.60 3.27 4.23 5.15

3 1.04 1.10 1.22 2.08 2.16 1.05 1.13 1.98 2.15 2.28 1.10 1.26 2.24 2.41 3.31 1.28 2.36 2.82 3.76 4.53

4 1.01 1.02 1.05 2.02 2.03 1.02 1.04 1.68 2.05 2.08 1.05 1.10 2.10 2.17 3.03 1.14 2.08 2.36 3.27 3.73

5 1.00 1.01 1.01 2.00 2.01 1.01 1.02 1.31 2.02 2.03 1.02 1.05 2.03 2.08 2.68 1.07 1.87 2.17 2.95 3.30

Table 5: The parameter value h of different control charts μ1

CUSUM

RLCUSUM1

RLCUSUM2

RLCUSUM3

FRLCUSUM3

1 2 3 3.5

5.05 5.52 5.53 5.19

5.05 5.44 5.65 5.24

5.06 5.44 5.65 5.24

5.14 5.56 5.74 5.31

5.19 5.59 5.78 5.35

GCUSUM 3.9

Table 6: The scope of application of five kinds of control charts Control charts CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 FRLCUSUM3

Small shift (δ ≤ 1) √ √ √

Medium shift (1 < δ < 1.5) √ √ √

√

√

Large shift (δ ≥ 1.5)

NOTE: δ is the calibrated magnitude of the shifts and δ = (μ − μ0 )/

21

√ √ √

 (1 − )σ02 + σ 2 .

Table 7: Notations Notations CUSUM RLCUSUM GLR GCUSUM FRLCUSUM ARL SPRT LLR RLLR 

Descriptions Cumulative sum control chart. Robust likelihood cumulative sum control chart. (RLCUSUM1, RLCUSUM2 and RLCUSUM3 respectively represent that the RLCUSUM control cahrt’s cumulative statistic is RLLR1, RLLR2 and RLLR3.) Generalized likelihood ratio. GLR-based CUSUM control chart. Fast initial response (FIR) RLCUSUM control chart. Average of run length. (ARL0 is on behalf of the in-control average of run length while ARL1 represents the out-of-control average of run length.) Sequential analysis ratio test. The log-likelihood ratio function. Robust log-likelihood ratio function. (RLLR1 , RLLR2 , RLLR3 respectively represent that the log-likelihood ratio cut off by horizontal ray, horizontal segment and oblique line.) Mathematical expression of LLR, RLLR1 , RLLR2 , RLLR3 .  Mathematical expression of cumulative sum, corresponding to the , ∗ , # ,  . Two horizontal axis of intersection points of horizontal ray and log likelihood ratio from left to right. Four horizontal axis of intersection points of horizontal segment and log likelihood ratio from left to right.

, ∗ , # ,   S, S ∗ , S # , S r1 , r2 p1 , p2 , p3 , p4 (x1 , y1 ), (x2 , y2 ), Four extreme points of contaminated normal distribution’s LLR from left to right. (x3 , y3 ), (x4 , y4 ) Four horizontal axis of intersection points of diagonal segment and log likelihood ratio from left to right. i1 , i2 , i3 , i4 k The slope of diagonal segment. Two horizontal axis of intersection points of horizontal ray and the part of monotonous decreasing log q1 , q2 likelihood ratio from left to right. a, b, c, d, e Five pints by which the interval [x1 , x2 ] are divided into equally four parts from left to right. Five parameters in likelihood ratio of contaminated normal distribution (1 − )N (μ0 , σ02 ) + N (μ0 , σ2 ), μ0 , μ1 , σ0 , σ,  where  is the degree of contamination and μ1 is the shifted value of mean. μ The mean of process. Head start value of fast initial response. γ h The control limit. λn Likelihood ratio function of samples. zi Log-likelihood ratio function of ith sample xi . ν The change point. Φ(·), φ(·) Cumulative distribution function and probability density function of standard normal distribution. f (·), F (·) Probability density function and cumulative distribution function. (x|μ1 ,σ0 )+f (x|μ1 ,σ) (x|μ1 ,σ0 )+F (x|μ1 ,σ) g(x) = log (1−)f − log (1−)F g(·) (1−)f (x|μ ,σ )+f (x|μ ,σ) (1−)F (x|μ ,σ )+F (x|μ ,σ) 0

0

0

0

22

0

0

ε=0.05

−1 −2

−2

0

10

20

−20

−10

0

x

x

ε=0.1

ε=0.3

10

20

10

20

1 0 −1 −2

−2

−1

zi

0

1

2

−10

2

−20

zi

0

zi

0 −1

zi

1

1

2

2

ε=0.01

−20

−10

0 x

10

20

−20

−10

0 x

4 2 −6 −4 −2 0

10

20

−20

−10

0

x

x

µ1=3

µ1=−3

10

20

10

20

4 2 0 −6 −4 −2

−6 −4 −2

0

zi

2

4

6

−10

6

−20

zi

0

zi

0 −6 −4 −2

zi

2

4

6

µ1=2

6

µ1=1

−20

−10

0 x

10

20

−20

−10

0 x

µ0=0,σ0=1,σ=2,ε=0.05

−10

4

6 0

10

20

−20

−10

0

10

x

µ0=0,σ0=1,σ=2,ε=0.25

µ0=0,σ0=1,σ=2,ε=0.5

20

4 2 0

0

2

g(x)

4

6

x

6

−20

g(x)

2 0

2

g(x)

4

6

µ1=1 µ1=2 µ1=3 µ1=3.5

0

g(x)

µ0=0,σ0=1,σ=2,ε=0.1

−20

−10

0 x

10

20

−20

−10

0 x

10

20

µ0=0,σ0=1,σ=3,ε=0.1 5

5

µ0=0,σ0=1,σ=3,ε=0.05

4 3 1 0 −1

−10

0

10

20

−20

−10

0

10

x

µ0=0,σ0=1,σ=3,ε=0.25

µ0=0,σ0=1,σ=3,ε=0.5

20

4 3 2 1 0 −1

−1

0

1

2

g(x)

3

4

5

x

5

−20

g(x)

2

g(x)

2 −1

0

1

g(x)

3

4

µ1=1 µ1=2 µ1=3 µ1=3.5

−20

−10

0 x

10

20

−20

−10

0 x

10

20

(a) LLR V.S. RLLR1

(b) LLR V.S. RLLR2

2

LLR RLLR3

2

LLR RLLR2

2

LLR RLLR1

(c) LLR V.S. RLLR3 (k=0.03)

p4=15.17 p3=2.25

r2=2.25

1 −1

zi

q2=(x3+x4)/2

0

1 0 −1

zi

0 −1

i3=1.50

i2=−0.50 i1=−11.07

q1=(x1+x2)/2

p2=−1.25

r1=−1.25,

−20

−10

0 x

10

20

−2

−2

p1=−14.17

−2

zi

1

i4=12.07

−20

−10

0 x

10

20

−20

−10

0 x

10

20

2.0

2.5

3.0

3.5

4.7 4.6 4.5

4.0

3.0

3.2

3.4

3.6

3.8

4.0

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.25

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.25

3.26 3.22

3.32

ARL1

3.30

3.36

µ

3.28

ARL1

4.4 4.3 4.2

k=0.01 k=0.02 k=0.03 k=0.04 k=0.05 k=0.06 k=0.07 k=0.08 k=0.09 k=0.1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.25

ARL1

6.5 7.0 7.5 8.0 8.5 9.0

ARL1

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

3.70

3.80

3.90 µ

4.00

3.70

3.80

3.90 µ

4.00

4.35 4.30 4.25

7.5 6.5

7.0

ARL1

8.0

k=0.01 k=0.02 k=0.03 k=0.04 k=0.05 k=0.06 k=0.07 k=0.08 k=0.09 k=0.1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.25

ARL1

8.5

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

2.0

2.5

3.0

3.5

4.0

3.5

3.7

3.8

3.9

4.0

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.25

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.25

3.28 3.24

3.26

3.34

ARL1

3.36

3.30

µ

3.32

ARL1

3.6

3.70

3.75

3.80 µ

3.85

3.70

3.75

3.80 µ

3.85

4.2

6.4

3.8

4.0

ARL1

6.8 6.6

2.6

2.8

3.0

3.2

3.4

3.5

3.7

3.8

3.9

4.0

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.25

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.25

3.33

ARL1

3.35

µ

3.31

ARL1

3.6

3.70

3.75

3.80 µ

3.85

3.20 3.22 3.24 3.26 3.28

ARL1

7.0

k=0.01 k=0.02 k=0.03 k=0.04 k=0.05 k=0.06 k=0.07 k=0.08 k=0.09 k=0.1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.25 4.4

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

3.70

3.75

3.80 µ

3.85

3.49 3.45

3.47

ARL1

5.50 5.45 5.40

ARL1

5.55

k=0.01 k=0.02 k=0.03 k=0.04 k=0.05 k=0.06 k=0.07 k=0.08 k=0.09 k=0.1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.25 3.51

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

3.5

3.6

3.7

3.8

3.9

4.0

3.70

3.75

3.80

3.85

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.25

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.25

3.08

3.12

3.16

ARL1

3.31 3.29

ARL1

3.33

3.20

µ

3.70

3.75

3.80 µ

3.85

3.70

3.75

3.80 µ

3.85

3.5

3.6

3.7

3.8

3.9

3.47

4.0

3.70

3.75

3.80

3.85

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.25

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.25

3.14 3.06

3.10

ARL1

3.31

3.33

3.18

µ

3.29

ARL1

3.45 3.43

ARL1

5.40 5.35

ARL1

5.45

k=0.01 k=0.02 k=0.03 k=0.04 k=0.05 k=0.06 k=0.07 k=0.08 k=0.09 k=0.1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.25 3.49

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

3.70

3.75

3.80 µ

3.85

3.70

3.75

3.80 µ

3.85

3.4 3.2 3.0 2.8

4.0

ARL1

4.5

5.0

a b c d e

2.6

3.5

ARL1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.05 3.6

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.05

2.5

3.0

3.5

4.0

2.5

2.7

2.8

2.9

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.05

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.05

3.0 2.8

2.8

ARL1

3.0

3.2

3.2

3.4

µ

2.6

2.6

ARL1

2.6

2.5

2.6

2.7 µ

2.8

2.9

2.5

2.6

2.7 µ

2.8

2.9

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.1 4.0 3.8 3.2

4.2

3.4

3.6

ARL1

5.0

5.4

a b c d e

4.6

ARL1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.1

2.5

3.0

3.5

4.0

2.5

2.6

2.7

2.8

2.9

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.1

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.1

3.4

ARL1

3.6

3.0

3.4 3.2

ARL1

3.8

3.8

4.0

µ

2.5

2.6

2.7 µ

2.8

2.9

2.5

2.6

2.7 µ

2.8

2.9

5.1 4.7

2.5

3.0

3.5

4.0

2.5

2.6

2.7

2.8

2.9

µ

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.25

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.25

4.6 4.2

ARL1

5.0

5.4

4.0 4.2 4.4 4.6 4.8 5.0

ARL1

4.9

6.0

ARL1

6.5

7.0

a b c d e

5.5

ARL1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.25 5.3

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

2.5

2.6

2.7 µ

2.8

2.9

2.5

2.6

2.7 µ

2.8

2.9

7.4

7.0

6.6

7.0

ARL1

9.0

10.0

a b c d e

8.0

ARL1

(b) µ0=0,µ1=2,σ0=1,σ=3,ε=0.5 7.8

(a) µ0=0,µ1=1,σ0=1,σ=3,ε=0.5

2.5

3.0

3.5

4.0

2.5

2.6

2.7

2.8

2.9

µ

(c) µ0=0,µ1=3,σ0=1,σ=3,ε=0.5

(d) µ0=0,µ1=3.5,σ0=1,σ=3,ε=0.5

7.5 6.5

7.0

ARL1

7.0 6.5

6.0

6.0

ARL1

8.0

7.5

8.5

µ

2.5

2.6

2.7 µ

2.8

2.9

2.5

2.6

2.7 µ

2.8

2.9

40 30 0

0

10

20

ARL1

30

CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 GCUSUM

10

5

10

15

20

0

10

15

µ0=0,µ1=1,σ0=1,σ=3,ε=0.25

µ0=0,µ1=1,σ0=1,σ=3,ε=0.5

20

30 10 0

10

20

ARL1

30

40

µ

0

ARL1

5

µ

40

0

20

ARL1

µ0=0,µ1=1,σ0=1,σ=3,ε=0.1

20

40

µ0=0,µ1=1,σ0=1,σ=3,ε=0.05

0

5

10 µ

15

20

0

5

10 µ

15

20

100 60 0 20

0

5

10

15

20

0

5

10

15

20

(a) µ0=0, µ1=3, ε=0.1, σ0=1, σ=3

(a) µ0=0, µ1=3.5, ε=0.1, σ0=1, σ=3

0

100

ARL1

50

200

mu

150

mu

0

ARL1

(b) µ0=0, µ1=2, ε=0.1, σ0=1, σ=3

ARL1

CUSUM RLCUSUM1 RLCUSUM2 RLCUSUM3 FRLCUSUM3 GCUSUM

10 20 30 40

ARL1

(a) µ0=0, µ1=1, ε=0.1, σ0=1, σ=3

0

5

10 mu

15

20

0

5

10 mu

15

20

(b) 0.4

12

(a)

0.2 0.0

0

2

0.1

4

x

Density

6

8

0.3

10

Real density Fitting density

0

20

40

60

Index

80

100

0

5 x

10

0 5

Sn 0 5

Sn

15

(b) RLCUSUM1

15

(a) CUSUM

40

60

80

100

0 8

20

40

60

index

Index

(c) RLCUSUM2

(d) RLCUSUM3

80

100

80

100

80

100

0 5

Sn 0 5

Sn

15

20

15

0 8

40

60

80

100

0 8

20

40

60

Index

Index

(e) FRLCUSUM3

(f) GCUSUM

0 5

Sn 0 5

Sn

15

20

15

0 8

03

20

40

60 Index

80

100

04

20

40

60 Index








The properties of RLCUSUM control schemes used to monitor the mean of contaminated normally distributed processes are evaluated. An optimal design procedure for RLCUSUM control schemes is given. In addition, several enhancements to RLCUSUM are considered. Compare three types of robust likelihood CUSUM control charts with conventional CUSUM and GCUSUM, and conclude their pros and cons. Fast initial response feature is included, that makes the RLCUSUM control schemes more sensitive to start-up cases. An application based on monitoring packing 250-gram containers is used to illustrate the implementation of RLCUSUM control schemes.

Acknowledgements This research is supported in part by the National Natural Science Foundation of China (11301323).