Inference for a simple step-stress model based on ordered ranked set sampling

Applied Mathematical Modelling 75 (2019) 23–36

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Inference for a simple step-stress model based on ordered ranked set sampling Mohammed S. Kotb a,b,∗, Mohammad Z. Raqab c,d a

Department of Mathematics, Al-Azhar University, Nasr City, Cairo 11884, Egypt Department of Mathematics, Al-Baha University, Saudi Arabia c Department of Mathematics, The University of Jordan, Amman 11942, Jordan d King Abdulaziz University, Jeddah, Saudi Arabia b

a r t i c l e

i n f o

Article history: Received 17 August 2018 Revised 10 May 2019 Accepted 15 May 2019 Available online 20 May 2019 Keywords: Bayes estimator Conjugate prior Exponential distribution Ordered ranked set sampling Step-stress accelerated life testing

a b s t r a c t In this paper, we considered the inference problem on simple step-stress accelerated life test data from one-parameter exponential distribution under type-I censored ordered ranked set sample with cumulative exposure model. The Bayesian estimators and credible intervals for the model parameters are developed and compared with the corresponding estimators based on simple random sampling. Two real data sets and numerical simulation evaluations are presented to illustrate all the results developed here. The simulation study indicated that the proposed Bayes estimators and credible intervals based on ordered ranked set sampling performed better than their counterparts using simple random sampling. © 2019 Published by Elsevier Inc.

1. Introduction Accelerated life test is a technique used to ensure a faster rate of failure and obtain adequate failure data within a reasonable testing time (in a short period). In this test, items are exposed to stress levels during the experiment such as temperature, pressure, vibration or voltage to get quick failures to infer the life characteristics under normal use. Accelerated life tests are performed more frequently than ordinary life tests because of increased reliability and advancement in technology, and these tests are reducing the burden in both testing time and expenditure for enterprises. Naturally, this type of tests allows us to improve the reliability of the components and identify the best one. For a detailed review on accelerated life test, one may refer to [1,2]. There are three common accelerated life test procedures according to the stress loading to the test units: constant stress where each item is tested under a constant level of stress [3]; step-stress testing in which the tested items are subjected to successively higher levels of stress for pre-assigned test times [4]; progressive step-stress in which stress is increased step-by-step over time under progressive censoring data [5]. We consider here a simple step-stress model under type-I censored ordered ranked set sample. The scheme of ranked set sampling (RSS) is of great interest as it has many applications in the area of industrial, social, biological and environmental studies. It is a cost-efficient alternative to simple random sampling (SRS) for obtaining a sample of observations and estimating the unknown population mean. The one-cycle ranked set sampling can be described as follows: randomly select

∗

Corresponding author. E-mail addresses: [email protected] (M.S. Kotb), [email protected] (M.Z. Raqab).

https://doi.org/10.1016/j.apm.2019.05.022 0307-904X/© 2019 Published by Elsevier Inc.

24

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

n2 units from the variable of interest and then allocate these units randomly into n sets, each of size n. The n units of each set are ranked visually or by any inexpensive method with respect to the variable of interest. By repeating the cycle q times yields a sample of size qn out of n2 q units (called as q-cycle RSS). Recently, Kotb and Raqab [6] discussed the Bayesian inference based on RSS data from the Rayleigh distribution. Several studies have been done in the RSS context to design new rank-based sampling techniques and construct an estimator that is more efficient than those based on SRS for estimating the location and scale parameters from various distributions. The ordered ranked set sampling (ORSS) is an alternative variation of RSS, which provides more accurate estimates of the population mean compared with the estimators obtained from RSS and SRS data [7]. In the context of ORSS, it is shown that the best linear unbiased estimators based on ORSS are more efficient than the ones based on RSS for Logistic, normal and two-parameter exponential distributions [7]. In this set-up, the resulting data are order statistics from independent and non-identically distributed random variables. The Bayesian prediction intervals using a proper general prior in the context of an ORSS from a certain class of exponential-type distributions are obtained by Kotb [8]. From the available literature, it can be seen that a lot of work has been done on step-stress accelerated life testing (SSALT) under different distributions. The exponential distribution is effectively incorporated into reliability studies. It can be used to model lifetime of product that fails at a constant rate. In practical applications, the failure rates of products might not be changed significantly with age. Although significant work has been done on SSALT and especially on Bayesian and non-Bayesian methods in the case of complete exponential data, the inference for SSALT under ORSS exponential data has not been considered. This motivates the present work to introduce the SSALT models and make Bayesian statistical inference under exponential ORSS data. In the SSALT models, the products are forced to fail faster than the ones under normal operating condition when there is time constraint on the duration of the experiment. These situations arise naturally in censoring data such as type-II right censoring, progressive censoring and hybrid censoring samples. The ORSS scheme is more efficient sampling method than the SRS, since it can be used to improve the cost efficiency of selecting sample units of an experiment. For this, the RSS scheme have been playing an important role in many practical aspect associated with applications in industrial, medical and environmental studies [9], since the interest lies in sampling designs that are cheaper than SRS and provided comparable estimates. The SSALT allows the experimenter to choose one or more stress factors in a life-testing experiment which may include humidity, temperature, voltage, load or any other factor that affects the life of the products. The most common model used to analyse these times-to-failure data is the cumulative exposure model. We consider here a simple step-stress model with only two stress levels in a situation where there is a time constraint on the duration of the experiment. The lifetime distributions at x1 and x2 are assumed to be exponential distribution with failure rates θ 1 and θ 2 , respectively. The rest of this paper is organized as follows. In Section 2, the basic set-up and model details are presented. The samplebased estimators using Bayesian approach under square error loss and general entropy loss functions are developed based on both SRS and ORSS data in Section 3. Two real data sets are analyzed in Section 4 to illustrate the results developed in Section 3. The performance of the Bayes estimators of the model parameters are evaluated in Section 5 using the mean square error and mean posterior risk. The performance of the corresponding credible intervals (CIs) are also shown in terms of the average length. Finally we conclude the paper in Section 6.

2. Model description Assume that the experiment starts with n2 number of identical units which are randomly allocated into n sets. The n units of each set are placed on a step-stress test at an initial stress level of x1 and the stress level is increased to x2 (x2 > x1 ) at a pre-fixed time τ . Now, let t j = (t j (1:n ) , tj(2:n) , , tj(n:n) ), j = 1, 2, . . . , n be the order statistics of n sets, each is based on SRS from the population of interest. Under type-I censoring, the experiment will terminate when all n products fail or censoring time (say, η) is reached. Further, let ni and wi − ni , i = 1, 2, . . . , n are number of units that occur before and after τ , respectively. The procedure of a one-cycle censored RSS can be described as follows: The m observed units within one-cycle RSS scheme, are ordered and denoted by (t1:n , . . . , tm:n) where t1:n ≤  ≤ tm:n ≤ η, with ti:n ≡ ti(i:n) , i = 1, 2, · · · , m. Generally, from Scheme 1, let t = t1:n , · · · , tn1 :n , tn1 +1:n , · · · , tm:n , where tn1 :n ≤ τ ≤ tn1 +1:n , tm:n ≤ η and m ≤ n, be the ordered observations of the lifetime T obtained from ORSS when all units fail or censoring time

Scheme 1. A ranked set sample design with sample size m (out of n).

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

25

η( > τ ) is reached. Under this ORSS scheme, n1 and m − n1 are the numbers of failures before and after time τ , respectively. It is clear that both ti:m and ti(i:m) have the same distribution which is the distribution of ith order statistic from a SRS. It is assumed that the lifetimes of the above units have an exponential distributions under both the stress levels, with the different scale parameters, say θ 1 and θ 2 , respectively. It is further assumed that the lifetime satisfies cumulative exposure distribution that takes the form of

⎧   ⎨G1 (t ) = 1 − exp − t , 0 < t < τ, θ1   F (t ) = ⎩G2 (s + t − τ ) = 1 − exp − s + t − τ , τ ≤ t < ∞, θ2

(1)

where s is the solution of G1 (τ ) = G2 (s ). The optimality issues of an SSALT under exponential cumulative exposure distribution was addressed in the literature [10]. To illustrate the proposed method of a one-and two-cycle censored RSS, we consider the real data set which proposed by Han and Kundu ([11], Table VII). The stress change time τ and the censoring time η are 5 and 6, respectively. The procedure of a one-cycle censored RSS can be described in Section 4. It is well known that the cumulative distribution function and probability density function of ti:m are, respectively,

Fi:m (ti ) = 1 −

i

m+ j−i

c j,i (1 − G(ti ) ) ,

0 < t < ∞,

(2)

j=1

and

fi:m (ti ) =

i−1

m+ j−i

c j,i (m )(1 − G(ti ) )

g(ti ),

0 < t < ∞,

(3)

j=0

cf. ([12], p.10), where

c j,i (m ) = (−1 ) j





i−1 ci,m , j

 m i

ci,m = i

and

c j,i = c j−1,i (m )/(m + j − i ).

3. Estimation of the model parameters In this section, we consider the estimation problem when the observed failure data are type-II censored under a simple SSALT. Bayes point and interval estimation of the model parameters, θ = (θ1 , θ2 ) based on one- and q-cycle ORSS are provided. 3.1. Estimation of parameters using one-cycle ORSS Let ti ≡ ti:n , i = 1, 2, · · · , m be the observed values of the lifetime T obtained from ORSS. The likelihood function of θ based on type-I censored ORSS is then given by

n1 m n    1 L1 (θ|t ) = fik (tk ) f ik ( s + tk − τ ) ( n − m )! P

k=1

k=n1 +1



1 − Fik (s + η − τ )

 

,

(4)

k=m+1

where P denotes the sum over all m! permutations (i1 , i2 , , im ) of (1, 2, , m). From [1] and Eq. (4), we readily write the likelihood function as

L1 (θ|t ) = where PerT =

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ T=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1

( n − m )!

 n P

PerT,

k=1 Ck,ik

f1 (t1 ) .. . f1 (tn1 ) f1 (tn∗1 +1 ) .. . ∗ f1 (tm ) 1 − F1 (η∗ )

(5)





denotes the permanent of a real matrix T = Ci,k of size n × n,

f2 (t1 ) .. . f2 (tn1 ) f2 (tn∗1 +1 ) .. . ∗ f2 (tm ) 1 − F2 (η∗ )

··· ..

.

··· ··· ..

.

··· ···

fn (t1 ) .. . fn (tn1 ) fn (tn∗1 +1 ) .. . ∗ fn (tm ) 1 − Fn (η∗ )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

}n − m rows

26

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

with ti∗ = (θ2 /θ1 )τ + ti − τ , η∗ = (θ2 /θ1 )τ + η − τ and ”}n − m rows” denotes that the row repeats n − m times. On substituting the cumulative distribution function and probability density function in (2) and (3), respectively, into (4), the likelihood function of θ = (θ1 , θ2 ) can be written as

ik −1 n1 i m k −1    n+−ik 1 n+−ik L1 (θ|t ) = c,ik (n )(1 − G1 (tk ) ) g1 (tk ) c,ik (n ) 1 − G2 (tk∗ ) g2 (tk∗ ) ( n − m )! k=1 =0

P

ik n 

×

k=n1 +1 =0



n+−ik

c,ik (1 − G2 (η∗ ) ) .

(6)

k=m+1 =1

Upon making use of (1) and the following identities n1 i k −1 

 (ik ) =

k=1 =0

i 1 −1 i 2 −1

···

j1 =0 j2 =0

ik n 

∇  ( ik ) =

im+1

in1 −1 n1 

im+2

···

νm+1 =1 νm+2 =1

k=m+1 =1

 jk ( ik ) ,

jn1 =0 k=1 in n 

∇νk (ik ),

νn =1 k=m+1

where i1 < i2 <  < in and ik , k = 1, 2, · · · , n are positive integers, we simplify the likelihood function (6) as



n1 1 ,m,n n  1 1 L1 ( θ ; t ) = h ¯ j ,ν ,v ( i ) ( n − m )! θ1 j ,ν ,v

P

k=1

m  k=n1 +1



1



exp −

θ2

u 1 ( t1 )

θ1

−

u 2 ( t2 )

θ2



,

(7)

where j = ( j1 , ..., jn1 ), ν = (νn1 +1 , ..., νm ), v = (vm+1 , ..., vn ), n 1 ,m,n j ,ν ,v

=

···

n1 

h ¯ j ,ν ,v ( i ) =



c jk ,ik (n )

i m −1

···

.

im+1

cνk ,ik (n )

···

n 

in

,

vn =1

 c˜vk ,ik ,

k=m+1

t2 = (tn1 +1 , tn1 +2 , · · · , tm ),

(n + jk − ik + 1 )tk +

m

( n + νk − i k + 1 ) τ +

k=n1 +1

m

im+2

νm =0 vm+1 =1 vm+2 =1



m 

k=1

u2 ( t2 ) =



k=n1 +1

t1 = (t1 , t2 , · · · , tn1 ), n1





k=1

u1 ( t1 ) =

.

jn1 =0 νn1 +1 =0 νn1 +2 =0

j1 =0 j2 =0

in1 +1 −1 in1 +2 −1

in1 −1

i 1 −1 i 2 −1

(n + νk − ik + 1 )(tk − τ ) +

k=n1 +1

n

( n + vk − ik ) τ ,

k=m+1 n

( n + v k − i k ) ( η − τ ).

k=m+1

To develop the Bayes estimators of θ 1 and θ 2 , we consider independent conjugate type of priors, π 1 (θ 1 ) and π 2 (θ 2 ). More specifically, we take π i (θ i ), i = 1, 2 as inverse gamma distribution (denoted as IG(γ i , β i )) with parameters γ i and β i . Then the joint prior density function of θ 1 and θ 2 has the form

   β β π θ ; δ = π1 (θ1 )π2 (θ2 ) ∝ θ1−γ1 −1 θ2−γ2 −1 exp − 1 − 2 , θ1 , θ2 > 0, θ1 θ2

(8)

where δ = (γ1 , γ2 , β1 , β2 ) is a vector of hyper-parameters with γ 1 , γ 2 > 0 and β 1 , β 2 > 0 being the shape and scale parameters, respectively. These hyper-parameters can be found using the arguments given in [13]. Here, let ϑi and σ i be the best guesses for the mean and standard deviation of θ i , i = 1, 2, respectively. Then,

ϑi =

βi

γi − 1

and

σi2 + ϑi2 =

βi2 . (γi − 1 )(γi − 2 )

Thus, the values of γ i and β i can be computed by the following equations:



ϑi γi = 2 + σi

2

and

βi = ϑi (γi − 1 ).

(9)

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

By combining (7) and (8), the posterior density of θ 1 and θ 2 is obtained to be



−(γ1 +n1 )−1 n 1 ,m,n



I−1 θ1

π1 θ|t =

θ2γ2 +m−n1 +1

= I−1

1 ,m,n n

j ,ν ,v

P

j ,ν ,v

P



h ¯ j,ν ,v (i ) exp −

u1 (t1 ) + β1

−

θ1

u2 (t2 ) + β2

27



θ2

h ¯ j,ν ,v (i )IG(γ1 + n1 , u1 (t1 ) + β1 )IG(γ2 + m − n1 , u2 (t2 ) + β2 ),

(10)

where

I = (n1 + γ1 )(m − n1 + γ2 )ψ (γ1 , γ2 ), with  (.) being the complete gamma function and 1 ,m,n n

ψ (γ1 , γ2 ) =

j ,ν ,v

P

h ¯ j ,ν ,v ( i )

1

×

n 1 + γ1

(u1 (t1 ) + β1 )

1

(u2 (t2 ) + β2 )m−n1 +γ2

.

It can be observed from (10) that the joint posterior density function is a mixture of inverse gamma density functions, IG(γ1 + n1 , u1 (t1 ) + β1 ) and IG(γ2 + m − n1 , u2 (t2 ) + β2 ). By (10), the Bayes estimators of θ 1 and θ 2 under square error loss function are given by

ψ (γ1 − 1, γ2 ) ψ (γ1 , γ2 − 1 ) and θ˜2 = . (γ1 + n1 − 1 )ψ (γ1 , γ2 ) (γ2 + m − n1 − 1 )ψ (γ1 , γ2 ) Further, the posterior second moment of θ 1 and θ 2 are given by   ψ (γ1 − 2, γ2 ) E θ12 |t = , (γ1 + n1 − 1 )(γ1 + n1 − 2 )ψ (γ1 , γ2 ) θ˜1 =

and

E



 θ22 |t =

ψ (γ1 , γ2 − 2 ) . (γ2 + m − n1 − 1 )(γ2 + m − n1 − 2 )ψ (γ1 , γ2 )

3.2. Estimation of the parameters using q-cycle ORSS





Let t j = t j1 , t j2 , · · · , t jn j , t j (n j +1 ) , · · · , t jm j , where t j1 ≤ t j2 ≤ · · · ≤ t jn j ≤ τ ≤ t j (n j +1 ) ≤ · · · ≤ t jm j ≤ η, j = 1, 2, · · · , q, be observed units based on q-cycle ORSS scheme from the same population. Here, we obtain the likelihood function as

L2 (θ|t ) =

q 

L1 (θ|tr )

r=1

=

q 

−1 ( n − m r )!

r=1

mr 

×

k=nr +1

r=1 P[r] j ,ν r ,vr r

 1

q ,mr ,n  nr



exp −

θ2

ur,1 (tr,1 )

θ1

t = (t1 , · · · , tq ), tr = (tr,1 , tr,2 ), (νr (nr +1) , · · · , νrmr ), vr = (vr (mr +1) , · · · , vrn ),

=

r=1 P[r] j ,ν r ,vr r nr ,mr ,n j r , ν r , vr

,m1 ,n n1 P[1] j1 ,ν 1 ,v1



···

nr

mr k=nr +1

θ1



,

θ2

(11) tr,2 = (tr (nr +1 ) , · · · , trmr ),

,



ir,mr −1

···



ir,mr +1

.



···

νr,mr =0 vr,mr +1 =1

(n + jr,k − ir,k + 1 )trk +

k=1

ur,2 (tr,2 ) =

.

−

ur,2 (tr,2 )

k=1



P [ q ] j q , ν q , vq

jr,nr =0 νr,nr +1 =0

jr,1 =0

ur,1 (tr,1 ) =



,mq ,n nq

ir,nr +1 −1

ir,nr −1

ir,1 −1

=

···

h ¯ j r , ν r , vr ( ir )

nr  1

tr,1 = (tr1 , · · · , trnr ),

where

q ,mr ,n  nr

mr

ir,n

,

vr,n =1

(n + νr,k − ir,k + 1 )τ +

k=nr +1

(n + νr,k − ir,k + 1 )(trk − τ ) +

n

(n + vr,k − ir,k )τ ,

k=mr +1 n k=mr +1

(n + vr,k − ir,k )(η − τ ).

jr = ( jr1 , · · · , jrnr ),

νr =

28

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

By making use of relation (1), we can write the likelihood function (11) as

q

L2 (θ|t ) ∝

P, j,ν ,v

q 



h ¯ j r , ν r , vr ( ir )

θ



¯ q −n¯ q ) −qn¯ q −q(m 1 2

θ

exp −

r=1

q 1

θ1

ur,1 (tr,1 ) −

r=1

q 1

θ2

 ur,2 (tr,2 ) ,

(12)

r=1

where q

q ,mr ,n  nr ,

=

P, j,ν ,v

r=1 P[r] j ,ν r ,vr r

P = (P[1], P[2], · · · , P[q] ), and

z¯q =

q 1 ( z1 + z2 + · · · + zq ). q i=1

It follows from (8) and (12) that the posterior density function:





π2 θ|t = J

q

−1

P, j,ν ,v

q 



h ¯ j r , ν r , vr ( ir )

θ1−(γ1 +qn¯ q )−1 θ2−(γ2 +qm¯ q −qn¯ q )−1

r=1



× exp −

1

q

θ1

r=1

 ur,1 (tr,1 ) + β1

−

1

q

θ2

r=1

 ur,2 (tr,2 ) + β2

,

(13)

where

¯ q − qn¯ q )ξ (γ1 , γ2 ), J = (γ1 + qn¯ q )(γ2 + qm

q

ξ (γ1 , γ2 ) =

P, j,ν ,v

q 

 h ¯ j r , ν r , vr ( ir )

r=1

q

γ1 +qn¯ q ur,1 (tr,1 ) + β1

r=1

q

γ2 +qm¯ q −qn¯ q . ur,2 (tr,2 ) + β2

r=1

It is clear that the joint posterior density function is a mixture of inverse gamma density functions, q  ¯ q − qn¯ q , qr=1 ur,2 (tr,2 ) + β2 ). It follows that the -the moment of the IG(γ1 + qn¯ q , r=1 ur,1 (tr,1 ) + β1 ) and IG(γ2 + qm posterior of θ 1 and θ 2 are

E

   (γ1 + qn¯ q −  ) ξ (γ1 − , γ2 ) × , θ1 | t = (γ1 + qn¯ q ) ξ (γ1 , γ2 )

E

   (γ2 + qm¯ q − qn¯ q −  ) ξ (γ1 , γ2 −  ) × . θ2 | t = (γ2 + qm¯ q − qn¯ q ) ξ (γ1 , γ2 )

and

The posterior reliability function of θ 1 and θ 2 given t are given, respectively, by

Rθ1 (|t ) =



∞



∞



0

= 1−

π2∗ (θ |t )dθ1 d θ2

ξ1∗ (γ1 , γ2 ) , (γ1 + qn¯ q )ξ (γ1 , γ2 )

(14)

ξ2∗ (γ1 , γ2 ) , (γ2 + qm¯ q − qn¯ q )ξ (γ1 , γ2 )

(15)

and

Rθ2 (|t ) = 1 − where

ξ (γ1 , γ2 ) = ∗ 1

q P, j,ν ,v

q 

 h ¯ j r , ν r , vr ( ir ) 

r=1 q r=1

γ1 + qn¯ q ,

γ1 +qn¯ q ur,1 (tr,1 ) + β1

q

  ur,1 (tr,1 ) + β1 /

r=1 q r=1

ur,2 (tr,2 ) + β2

γ2 +qm¯ q −qn¯ q ,

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

ξ (γ1 , γ2 ) = ∗ 2

q

q 

 h ¯ j r , ν r , vr ( ir ) 

γ2 + qm¯ q − qn¯ q ,

r=1

P, j,ν ,v

q

γ1 +qn¯ q ur,1 (tr,1 ) + β1

r=1

q

  ur,2 (tr,2 ) + β2 /

r=1 q

29

γ2 +qm¯ q −qn¯ q

,

ur,2 (tr,2 ) + β2

r=1

and  (x, y) is the incomplete gamma function. Now, we estimate θ 1 and θ 2 by considering a general entropy loss function which is proposed by Calabria and Pulcini [14]. A general entropy loss function is a suitable alternative to modified LINEX loss function [15]. Hence, the Bayes estimator based on the general entropy loss function, θ˜iGE of θ i is given by

  −1/λ θ˜iGE = E θi−λ |t , i = 1, 2, 

(16)



provided that E θi−λ |t exists and is finite. From (13) and (16), the Bayes estimators of θ 1 and θ 2 under general entropy loss function can be expressed, respectively, as



θ˜1GE = and

 θ˜2GE =

(γ1 + qn¯ q + λ ) ξ (γ1 + λ, γ2 ) × (γ1 + qn¯ q ) ξ (γ1 , γ2 )

−1/λ

,

(γ2 + qm¯ q − qn¯ q + λ ) ξ (γ1 , γ2 + λ ) × (γ2 + qm¯ q − qn¯ q ) ξ (γ1 , γ2 )

−1/λ .

For λ = −1, the general entropy loss function is square loss function, while the weighted square loss function obtained by setting λ = 1 and therefore almost symmetric. For λ = −2, the general entropy loss function is referred to as the precautionary loss function which is an asymmetric loss function. For λ < 0, a negative error has a more serious effect than a positive error and for λ > 0, a positive error has a more serious effect than a negative error. 3.3. Interval estimation Here we aim at developing two-sided Bayes credible intervals (CIs) as well as highest posterior density CIs for the model parameters, θ 1 and θ 2 . A symmetric 100(1 − α )% two-sided Bayes CI of θ 1 can be obtained by solving the following two equations:

1 − R θ1 ( θ 1 L | t ) = and

Rθ1 (θ1U |t ) =

α 2

α 2

,

,

where Rθ1 (.|t ) is the posterior reliability function of θ 1 given in (14). Similarly, a symmetric 100(1 − α )% two-sided Bayes CI of θ 2 can be obtained by solving the following two equations:

1 − R θ2 ( θ 2 L | t ) =

α 2

and Rθ2 (θ2U |t ) =

α 2

,

where Rθ2 (.|t ) is the posterior reliability function of θ 2 given in (15). Hence, the Bayes estimator of θ j , θ˜ j , verifies that θ jL ≤ θ˜ j ≤ θ jU , j = 1, 2. It is important to point out that the previous CIs do not specify whether the values of θ j ( j = 1, 2 ) within these intervals have highest probability than that of the values outside the intervals. For this, it is more desirable to have a CI with the highest posterior density (HPD). For any probability content, 1 − α , the HPD interval is of the shortest width and the posterior density for every point outside the interval is less than that for every point inside the interval. In this context, the HPD method requires finding two bounds, CjL and CjU for θ j (see, for example, [16], p. 110) satisfying the following requirements:



∞ 0



C jU

C jL

π2 (θ j , θk |t )dθ j dθk = 1 − α , and



∞

0

π2 (C jL , θk |t )dθk =

 0

∞

π2 (C jU , θk |t )d θk ,

where j, k = 1, 2 and j = k. From the above equations and after some algebraic computation, the 100(1 − α )% HPD CI for θ j , can be solved numerically from the following equations:









Rθ j C jL |t − Rθ j C jU |t = 1 − α ,



and

C jL C jU

ψ j +1

  φ j C jL  , = φ j C jU

j = 1, 2,

30

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

where

 ψ1 = γ1 + q n¯ q , ψ2 = γ2 + q m¯ q − qn¯ q ,

and



q

φ j (ρ ) =

P, j,ν ,v

q 

 exp − h ¯ j r , ν r , vr ( ir )

r=1

q

  ur, j (tr, j ) + β j /ρ

r=1 q

ψk

,

j = k.

ur,k (tr,k ) + βk

r=1

3.4. Estimation of the parameters using q-cycle SRS Let t = (tr1 , tr2 , · · · , trwr ), where tr1 ≤ tr2 ≤ · · · ≤ tnr ≤ τ ≤ tnr +1 ≤ · · · ≤ trwr ≤ η, r = 1, 2, · · · , q be the q− cycle SRSs obtained from the given model. Then, the likelihood function of these observed samples is given by q 

L3 (θ |t ) =

r=1

θ

∝ where n¯ = (1/q )

nr  n! g1 (trk ) ( n − w r )! k=1

−qn¯ q −q(ω ¯ q −n¯ q ) 1 2

θ

q r=1

nr



exp −

nr , ω ¯ = (1/q )



Qq

θ1

q r=1

trk + q(n − n¯ q )τ

r=1 k=1

and

( ) (1 − G2 (η∗ ) )n−wr

∗ g2 trk

k=nr +1

−



Qq∗

,

θ2

(17)

wr ,

q

Qq =



wr 

Qq∗ =

q mr

(trk − τ ) + q(n − ω¯ q )(η − τ ).

r=1 k=nr +1

From (8) and (17), the joint posterior density of θ 1 and θ 2 given t is

π3 (θ |t ) =

 γ2 +qω¯ q −qn¯ q ¯ (Qq + β1 )γ1 +qnq Qq∗ + β2 θ1−(γ1 +qn¯ q )−1 (γ1 + qn¯ q )(γ2 + qω¯ q − qn¯ q )  Qq∗ + β2 Qq + β1 − ( γ +qω ¯ q −qn¯ q )−1 ×θ 2 2 exp − − . θ1 θ2

Consequently, under a square loss function, one can easily show that

 ∗     (Qq + β1 ) (γ1 + qn¯ q −  )   Qq + β2 (γ2 + qω ¯ q − qn¯ q −  ) E θ1 = and E θ2 = . (γ1 + qn¯ q ) (γ2 + qω¯ q − qn¯ q )

The posterior reliability function of θ 1 and θ 2 given t are given, respectively, by

R¯ θ1 (|t ) = 1 − and

R¯ θ2 (|t ) = 1 −

(γ1 + q n¯ q , (Qq + β1 )/ ) , (γ1 + q n¯ q )

     γ2 + q ω¯ q − qn¯ q , Qq∗ + β2 / . (γ2 + q ω¯ q − q n¯ q )

4. Data analysis Here, we present the analysis of two real data sets with exponential fitting distribution to illustrate the methods described in Section 3. All the computations are conducted using Mathematica software. Example 1 (Electronic Device Data Set). The first real data were presented in [11, Table VII]. A simple step-stress test was conducted under a time constraint, which has two dominant failure modes (capacitor failure and controller failure) for assessing the reliability characteristics of a solar lighting device. In this case, temperature is the stress factor whose level was changed during the test from 293 K to 353 K with the typical operating temperature at 293 K. The stress change time τ and censoring time η are 5 and 6 (in hundred hours), respectively. It is assumed that, at any constant temperature, lifetime of the device follows exponential distribution. The data are displayed in Table 1. Example 2 (SUAV Reliability Data Set). The second real data were presented in [17, Table 7]. A simple step-stress test was conducted under time constraint to assess the reliability characteristics of a prototype small/micro unmanned aerial vehicle

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

31

Table 1 Data set from n = 35 prototypes of a solar lighting device on a simple step-stress test with two failure modes, τ = 5 and η = 6. Temperature Level 1 (before τ = 5)

0.140 2.660

0.783 2.674

1.324 2.725

1.582 3.085

1.716 3.924

1.794 4.396

1.883 4.612

2.293 4.892

Temperature Level 2 (after τ = 5)

5.002 5.305

5.022 5.337

5.082 5.407

5.112 5.408

5.147 5.445

5.238 5.483

5.244 5.717

5.247

Table 2 Data set from n = 23 prototypes of a SUAV for a simple step-stress test with two complementary risks, τ = 15 and η = 20. Wind speed level 1 (before τ = 15)

2.365(1) 11.416(1)

3.467(2) 11.789(1)

5.386(2) 12.039(2)

7.714(2) 14.928(1)

9.578(1) 14.938(2)

9.683(2)

Wind speed level 2 (after τ = 15)

15.325(2) 17.803(1)

15.781(2) 19.578(2)

16.105(1)

16.362(2)

17.178(2)

17.366(1)

Fig. 1. The empirical and fitted survival function.

(SUAV) for civil applications, including aerial forest surveys, fire surveillance, and disaster management. The SUAV is an aircraft under autonomous control by an automatic system under remote control by an operator on the ground. The SUAV was equipped with dual propulsion systems that operated independently. When both systems fail, the SUAV loses its flying altitude before completing its mission. Failure of either propulsion system does not cause the SUAV to malfunction, but the failure time is digitally recorded. In this case, the wind speed is the stress factor whose level was changed during the test. The stress change time τ and censoring time η are 15 and 20, respectively. The data are presented in Table 2. From this data set, we have n1 = 11 and m = 19. We first check whether the exponential distribution is appropriate for the both data sets in Examples 1 and 2 by using the Kolmogorov-Smirnov distance and Kaplan–Meier [18] estimator. Figs. 1 and 2 show, respectively, the KolmogorovSmirnov distance and the P-P plot of the Kaplan–Meier estimator versus the fitted exponential survival function for the given data. Visually, the depicted points for the fitted survival function are very near the 45o line, indicating very good fit. By using the procedure of RSS (see Scheme 1) and for fixed values of the stress change time τ and η, the procedure of a one- and two-cycle ORSS can be described in Schemes 2 and 3 for Examples 1 and 2, respectively. From Scheme 2, we will select the units (0.140, 2.293, 5.112, 5.445) and (0.783, 3.924, 5.238, 5.717) from ORSS when q = 1 and 2, respectively. By using Eq. (9), we have (γ1 , β1 ) = (148.347, 117.430 ) and (γ2 , β2 ) = (84.216, 61.315 ). Similarly, from Scheme 3, we will select the units (2.365, 7.714, 15.325, 16.362) and (3.467, 11.416, 17.803) when q = 1 and 2, respectively, with (γ1 , β1 ) = (93.590, 2437.740 ) and (γ2 , β2 ) = (55.173, 190.387 ). Hence, under square error loss and general entropy loss functions, the different Bayes estimates and their corresponding CIs of θ 1 and θ 2 are computed for both data sets and displayed in Table 3. For better understanding, the relative efficiencies of the Bayes estimate (say, ϕ˜ ) and CI of ϕ under ORSS scheme with respect to the corresponding estimate and CI under SRS scheme are computed, respectively, as

RE1 =

P R(ϕ˜ ORSS ) width of CI under ORSS , and RE2 = , width of CI under SRS P R(ϕ˜ SRS )

where PR is the posterior risk (posterior variance). It is clearly observed from Table 3 that the Bayes estimates under ORSS scheme perform well when compared to the respective estimates under SRS scheme. Further, the HPD CIs are shorter than

32

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

Fig. 2. P-P plot of Kaplan–Meier estimator versus fitted survival function.

Scheme 2. ORSS design, using illustrative Example 1.

Scheme 3. ORSS design, using illustrative Example 2.

that of Bayes CIs and both these CIs under ORSS have higher performances when comparing with their counterparts under SRS. To check the effect of the prior distribution, the Bayes estimates and CIs are also computed based on three-hundred different random values of the hyper-parameters and the results are presented in Table 3. It is evident that the performances of Bayes estimates under the arguments given in [13] are better than their corresponding Bayes estimates under the prior based on random values of hyper-parameters

5. Simulation study Here we perform a Monte Carlo simulation study to examine the performances of the Bayes estimates for SSALT under cumulative exposure model based on ORSS. Our goal here is to examine the performances of the sample-based estimates of the model parameters and the so obtained CIs. We compare the performances of the Bayes estimates in terms of two different optimality criteria (mean square errors and minimum posterior risks) and assess the Bayes and HPD CIs in terms of the average lengths. Further, we check the efficiencies of the Bayes estimates, Bayes CIs and HPD CIs based on ORSS with respect to those estimates based on SRS. For a particular n(= 4, 6, 8 ), stress change time τ and a censoring time η, we generate a type-I censored sample from the exponential distribution when q = 1, 2, 6. For these choices, the average numbers of failure observed before and after τ

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

33

Table 3 Estimates of θ 1 and θ 2 , posterior risks of the Bayes estimates, relative efficiencies of the Bayes estimates and average lengths of 95% CIs for unknown parameters of q−cycle.

Data Set

q

θ˜1

PR(θ˜1 )

RE1 (θ˜1 )

θ˜iGE

θ˜2

PR (θ˜2 )

RE1 (θ˜2 )

λ = −2

Bayes CI

HPD CI

Bayes CI

HPD CI

λ=2

AL

AL

RE2

RE2

7.8801 0.7260 9.0826 0.8153

2.5313 0.3136 2.4157 0.2350

2.4867 0.2607 2.1508 0.1934

0.5818 0.8617 0.7618 0.9722

0.6409 0.7371 0.3948 0.8675

26.3700 3.5162 26.4031 3.6007

25.8415 3.3959 25.8917 3.4811

10.4114 1.8218 10.2462 1.7857

10.3186 1.7942 10.2336 1.6982

0.6635 0.9666 0.6689 0.9641

0.7193 0.9637 0.7296 0.8635

3.0447 0.6257 4.8841 0.5819

2.7110 0.4880 4.4009 0.4997

2.7294 0.3703 2.6614 3.5280

2.5271 0.3510 2.2973 0.2551

0.6287 0.9670 0.6137 0.8831

0.8104 0.8807 0.8009 0.8455

8.3329 1.1977 12.8371 1.7201

7.3548 0.9485 11.4678 1.4588

12.9343 2.3864 11.8876 2.0947

12.4181 1.9302 11.3508 1.7925

0.6829 0.9370 0.7084 0.9138

0.7011 0.8019 0.7409 0.8924

Based on the arguments given in Sun and Berger (1994) 1 1 7.9575 0.4178 0.3382 7.9837 0.7386 0.0064 0.7416 0.7429 2 9.2657 0.2987 0.1929 9.3195 0.8389 0.0017 0.1290 0.8399 2

1 2

26.2355 3.4852 26.2730 3.5699

7.0761 0.2172 6.8524 0.1996

0.4382 0.9321 0.4454 0.9273

Based on random values of the hyper-parameters 1 1 2.9504 0.8344 0.9381 0.5812 0.0986 0.9924 2 4.7487 0.7251 0.7068 0.6575 0.0458 0.6394 2

1 2

are

8.0515 1.1174 12.4522 1.6429

1000 n¯ 1 =

n 1i ¯ = and m 10 0 0 i=1

7.3738 0.3447 5.8101 0.3080

0.7331 0.9805 0.7309 0.9376

1000

i=1 mi , 10 0 0

respectively. For computing the Bayes estimators, we assume the prior densities π 1 (θ 1 ) and π 2 (θ 2 ) of θ 1 and θ 2 , respectively, given in (8) with the following hyper-parameters: γ1 = 4.3, β1 = 5.0, γ2 = 4.0 and β2 = 6.0. Based on this informative prior, we generate the values θ1 = 1.5143 and θ2 = 2.0 0 05, and then we have obtained s = (θ2 /θ1 )τ , for fixed values of τ . In each case, we compute the Bayes estimates of θ 1 and θ 2 . We replicate the process 10 0 0 times and compute the averages, mean square errors and minimum posterior risks of Bayes estimators as well as average lengths of Bayes and HPD CIs. The following algorithm describes the steps to generate the sample-based estimates, CIs and the corresponding optimality criteria for the assessment. MC Simulation Algorithm: Step 1: Generate n independent uniform U(0, 1) variates (say, U1 , U2 , . . . , Un ). Step 2: For given values of the change time τ and censoring time η, find ni and mi such that

Uni :n ≤ 1 − e

− θτ

1

< Uni +:n , Umi :n ≤ 1 − e

− s+θη−τ 2

< Umi +1:n .

Step 3: Using τ , η and mi , we generate m random samples, ti(1:n) , , ti(ni :n ) , ti(ni +1:n ) , · · · , ti(mi :n ) , i = 1, 2, · · · , mi , each of size mi by using the transformation:

 −1 θ1 log 1 − U j:n , 1 ≤ j ≤ ni , ti( j:n ) =  −1 θ2 log 1 − U j:n + τ − s, ni + 1 ≤ j ≤ mi .

(18)

Step 4: Obtain m ordered observations, t1:n ≤ · · · ≤ tn1 :n ≤ tn1 +1:n ≤ · · · ≤ tm:n ≤ η, from RSS sample Step 5: Repeat steps (2–4) q runs to obtain q-cycle ORSS data. Based on the iterated samples (say, n∗ = 10 0 0), the average Bayes estimator, minimum posterior risk and mean square error of ϕ , are, respectively, ∗

∗

n n 1 1 ϕ˜¯ = ∗ ϕ˜ j , MPR(ϕ˜ ) = ∗ P R(ϕ˜ j ), MSE (ϕ˜ j ) = n n j=1

j=1

n ∗

i=1



ϕ˜ j − ϕ˜¯

2

n∗

,

Let us define the efficiencies of the Bayes estimator ϕ˜ of ϕ and CI of ϕ as follows:

RE3 =

n ˜¯ 1 |ϕ˜ j − ϕ| , n∗ |ϕ˜ jGE − ϕ| ˜¯ j=1

RE4 =

ORSS n 1 MSE (ϕ˜ j ) , n∗ MSE (ϕ˜ SRS ) j j=1

RE5 =

ORSS n 1 P R(ϕ˜ j ) , n∗ P R(ϕ˜ SRS ) j j=1

RE6 =

Average length of CI under ORSS . Average length of CI under SRS

∗

∗

∗

34

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36 Table 4 The averages, mean square errors and minimum posterior risks of the Bayes estimates and average lengths of 95% CIs for unknown parameters of q−cycle.

τ

θ˜ 1

MSE (θ˜1 )

MPR(θ˜1 )

MSE (θ˜2 )

MPR(θ˜2 )

RE3 (θ˜i )

Bayes CI

HPD CI

q

n

η

¯ m

θ˜ 2

λ = −2

λ=2

AL

AL

1

4

0.1 1.8 0.9 1.9

1.0379 2.9372 1.9750 3.0381

1.1893 1.5848 1.3575 1.2352

0.5119 0.6336 0.3115 0.9806

0.4117 0.4312 0.2725 0.8502

1.1051 1.1720 1.5090 3.2093

0.5139 0.5713 0.4916 0.2776

2.3572 2.5157 1.9129 3.3080

1.9936 2.2044 1.8237 2.8039

6

0.1 1.1 0.5 1.8

1.1059 3.0753 1.9910 4.0155

1.2487 1.6035 1.3547 1.8386

0.5086 0.6075 0.2877 0.7467

0.4014 0.5628 0.2269 0.7054

1.1472 1.3512 1.3247 1.7163

0.4509 0.3729 0.4007 0.3254

2.3891 2.8591 1.6485 2.9307

2.0623 2.5465 1.5569 2.6785

8

0.35 1.1 1.05 1.3

1.9902 4.0176 4.0110 5.9998

1.3011 1.8215 1.3981 1.7355

0.3112 0.7637 0.1705 0.6137

0.2884 0.7500 0.1432 0.5977

1.4817 2.2918 1.3956 1.9276

0.3608 0.2142 0.2670 0.2532

2.0036 3.2239 1.6101 3.2668

1.7814 2.7842 1.5085 2.9785

4

0.1 1.8 0.9 1.9

1.0837 3.0291 2.0119 2.9918

1.2012 1.6409 1.3956 1.4658

0.4652 0.5132 0.2158 0.7766

0.2169 0.2266 0.1874 0.6756

1.0360 1.0293 1.1800 1.1610

0.4715 0.5253 0.4353 0.2438

1.7299 1.7570 1.6918 2.9957

1.5199 1.6522 1.4960 2.6288

6

0.1 1.1 0.5 1.8

1.0931 3.1082 2.0639 4.0017

1.3041 1.7976 1.4128 1.8174

0.4118 0.3766 0.1639 0.4689

0.2515 0.3226 0.1080 0.4132

1.0546 1.0980 1.0792 1.2287

0.6416 0.4857 0.6733 0.4402

1.9143 2.1970 1.2781 2.1424

1.6886 2.0224 1.1193 2.0328

8

0.35 1.1 1.05 1.3

2.0375 4.0037 4.0238 6.0017

1.3946 1.8804 1.5231 1.8325

0.1891 0.4650 0.0893 0.4730

0.1520 0.4443 0.0766 0.4069

1.0940 1.2278 1.1760 1.1465

0.5344 0.3111 0.9687 0.5178

1.8942 2.5208 1.0665 2.5189

1.4925 2.3122 1.1194 2.0941

4

0.1 1.8 0.9 1.9

1.0191 3.0371 2.1094 3.0107

1.3490 1.7351 1.4993 1.6205

0.3018 0.3188 0.0784 0.3951

0.0410 0.0752 0.0596 0.2468

1.0044 1.0067 1.0430 1.0353

0.9014 0.8705 0.8042 0.7074

0.9797 1.2582 1.1185 1.9173

0.9380 1.2199 1.0024 1.7670

6

0.1 1.1 0.5 1.8

1.0391 3.0101 1.9637 4.0811

1.463 1.8984 1.4639 1.8742

0.2852 0.1128 0.0919 0.1240

0.0590 0.1195 0.0283 0.1362

1.0119 1.0486 1.0114 1.0322

0.8710 0.6452 0.8635 0.7110

1.0670 1.3708 0.9181 1.3063

0.9903 1.3610 0.6907 1.3567

8

0.35 1.1 1.05 1.3

2.0055 4.1003 4.0056 6.1283

1.4901 1.9217 1.5471 1.9736

0.0913 0.1010 0.0246 0.0770

0.0429 0.1469 0.0208 0.1124

1.0246 1.0429 1.0864 1.0755

0.7671 0.3969 0.4830 0.5065

0.9589 1.8506 0.7391 1.2552

0.9270 1.5764 0.6388 1.2067

2

6

∗

n¯ 1

MSE and MPR denote the mean square error and minimum posterior risk, respectively.

Clearly if REi (i = 4, 5, 6 ) < 1, then the Bayes estimator of ϕ and corresponding CI based on ORSS outperform the Bayes estimator and the corresponding CI based on SRS, respectively. If RE3 > 1, then the Bayes estimator under general entropy loss function outperforms the corresponding Bayes estimator under square error loss function. The Bayes estimators and average lengths of CIs are displayed in Tables 4 and 5 for different choices of n, τ , and η. From the numerical results presented in Tables 4 and 5, we may report the following points:

1. For fixed values of η, if the stress change time τ is increased (number failure units before τ is increased), the mean square errors, minimum posterior risks and the average lengths decrease with θ 1 while this observation is not clear for θ 2. 2. For fixed values of τ , the mean square errors, minimum posterior risks and the average lengths decrease with θ 1 and θ 2 when the censoring value η is increased (i.e m − n1 is increased). 3. For all considered censoring schemes, the mean square errors, minimum posterior risks and the average lengths of CIs obtained for θ 1 is less than those for θ 2 . 4. Based on the values of relative efficiencies, it can be observed that the Bayes estimators under general entropy loss function when λ = −2 might compete the Bayes estimators under square error loss function. 5. It is evident that ORSS sampling design is a quite efficient than SRS sampling design based on the same sample size n. Moreover, the efficiency based on ORSS data increases as q (q−cycle) increases for all cases considered in this study. 6. Finally, the HPD CIs are shorter than two-sided Bayes CIs.

M.S. Kotb and M.Z. Raqab / Applied Mathematical Modelling 75 (2019) 23–36

35

Table 5 Relative efficiencies the Bayes estimators, Bayes CIs and HPD CIs of θ 1 and θ 2 based on ORSS with respect to SRS.

τ

n¯ 1

RE4 (θ˜1 )

RE5 (θ˜1 )

Bayes CI

HPD CI

q

n

η

¯ m

RE4 (θ˜2 )

RE5 (θ˜2 )

RE6 (θ˜1 )

RE6 (θ˜2 )

RE6 (θ˜1 )

RE6 (θ˜2 )

1

4

0.1 1.8 1.0 1.9 0.1 1.1 0.5 1.8 0.35 1.1 0.35 1.1 0.1 1.8 1.0 1.9 0.1 1.1 0.5 1.8 0.35 1.1 1.05 1.3 0.1 1.8 1.0 1.9 0.1 1.1 0.5 1.8 0.35 1.1 0.35 1.1

1.0379 2.9372 2.0990 3.0400 1.1059 3.0753 1.9910 4.0155 1.9902 4.0176 4.0110 5.9998 1.0837 3.0291 2.0119 2.9918 1.0931 3.1082 2.0639 4.0017 2.0375 4.0037 4.0238 6.0017 1.0191 3.0371 2.1094 3.0107 1.0391 3.0101 1.9637 4.0811 2.0055 4.1003 4.0056 6.1283

0.9576 0.7182 0.6369 0.9773 0.9272 0.6134 0.6092 0.8256 0.6656 0.8001 0.3658 0.7020 0.9728 0.8663 0.6579 0.9172 0.9580 0.5753 0.6421 0.7846 0.6050 0.7446 0.4019 0.8145 0.9706 0.9892 0.6391 0.9907 0.9838 0.6160 0.6109 0.7378 0.5236 0.7062 0.3483 0.7025

0.8774 0.5138 0.5653 0.9394 0.8747 0.5710 0.4883 0.8137 0.6300 0.7908 0.3295 0.6917 0.8066 0.4412 0.5952 0.8526 0.8151 0.4953 0.3723 0.7508 0.5261 0.7206 0.3501 0.7706 0.7120 0.4301 0.5485 0.7620 0.7196 0.4680 0.3272 0.6732 0.4599 0.6301 0.3182 0.6822

0.9504

0.7388

0.9361

0.7487

0.7514

0.9700

0.8147

0.9723

0.9268

0.7956

0.9316

0.8117

0.6549

0.8407

0.7085

0.8283

0.7972

0.9053

0.8137

0.8924

0.7011

0.9322

0.7474

0.9361

0.8997

0.6380

0.8980

0.6907

0.8164

0.9255

0.7903

0.9271

0.9316

0.7139

0.9357

0.7290

0.6517

0.8150

0.6823

0.8196

0.9303

0.8505

0.7994

0.8270

0.5926

0.8479

0.6847

0.7968

0.8580

0.6058

0.7637

0.6354

0.5907

0.7709

0.6336

0.7231

0.7155

0.6016

0.7519

0.6297

0.5418

0.7335

0.5232

0.7107

0.6396

0.7172

0.6814

0.6034

0.5073

0.7934

0.5827

0.6713

6

8

2

4

6

8

6

4

6

8

6. Conclusions In this article, based on the simple step-stress model from exponential distribution, the estimation of the unknown parameters and the corresponding credible intervals using both symmetric and asymmetric loss functions based on ordered ranked set sampling are discussed, from a Bayesian viewpoint. For different sample sizes, stress change time τ and censoring time η, these Bayes estimators were compared with the corresponding ones based on simple random sampling via numerical simulation in the sense of mean square errors and minimum posterior risks. The credible intervals were compared in terms of the average lengths. Further, the Bayes estimators under general entropy loss function compete the corresponding estimators under square error loss function well. It is observed that overall, the performances of Bayes estimators and credible intervals for θ 1 and θ 2 improve when the number of failures before and after time τ gets large. The relative efficiencies of the estimates for step-stress accelerated life testing model based on ordered ranked set sampling data increase as q (q-cycle) increases since the sampling scheme gets more informative when q tends to be large. Clearly, based on the numerical computations, the performances of step-stress accelerated life testing under ordered ranked set sampling scheme do better when compared to the step-stress accelerated life testing under simple random sampling scheme for all the considered cases. Throughout the study, we have mainly restricted our attention for analysing the step-stress accelerated life testing using ordered ranked set sampling design as one of rank-based sampling designs, but the analysis can be extended to other rank-based designs as well. More investigation is needed along this line.

Acknowledgments We would like to appreciate the constructive comments by an associate editor and two anonymous referees which improved the quality and the presentation of our results.

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