Statistical Inference for the Rayleigh distribution under ...

Applied Mathematical Modelling 38 (2014) 974–982

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

Statistical Inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal Sanku Dey a, Tanujit Dey b,⇑ a b

Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India Department of Mathematics, The College of William & Mary, Williamsburg, VA, USA

a r t i c l e

i n f o

Article history: Received 12 November 2012 Received in revised form 24 May 2013 Accepted 4 July 2013 Available online 25 July 2013 Keywords: Rayleigh distribution Maximum likelihood estimator Bayes estimator Binomial censoring scheme Type II progressive censoring Expected termination point

a b s t r a c t This paper takes into account the estimation for the unknown parameter of the Rayleigh distribution under Type II progressive censoring with binomial removals, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. The expected termination point to complete the censoring test is computed and analyzed under binomial censoring scheme. Numerical examples are given to illustrate the approach by means of Monte Carlo simulation. A real life data set is used for illustrative purposes in conclusion. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Rayleigh distribution has been widely used in reliability theory and survival analysis, as its failure rate is a linear function of time. The reliability function of Rayleigh distribution decreases at a much higher rate than the reliability function of exponential distribution. The origin and other aspects of this distribution can be found in [1]. This distribution plays an important role in real life applications since it relates to a number of distributions such as generalized extreme value, Weibull and Chi-square distributions. In statistical literature a good amount of work has been devoted to Rayleigh distribution. Several authors such as [2–4] and the references cited therein have carried out extensive studies as relate to the estimation, prediction and several other inferences with respect to Rayleigh distribution. Although statistical inference for life time distributions based on complete sample ([5]) is of great interest; for last few decades, there has been an overwhelming consideration for censored samples evolving from numerous life time distributions. To name a few, [6] studied optimal designing of stepstress partially accelerated life tests under Type-I and Type-II censoring scheme; [7] obtained E-Bayes estimates for the parameters and reliability function of the Burr Type XII distribution based on Type-II censored samples; [8] studied Weibull distribution under step-stress partially accelerated life tests with hybrid censored data. The Rayleigh distribution has proven to be effective while studying data censoring schemes, common in most life testing experiments. Among the different censoring schemes, the progressive Type-II censoring scheme has most widely been used particularly in reliability analysis and survival analysis. It is preferred to traditional Type-II censoring scheme. Progressive censoring is useful in both industrial life testing applications and clinical settings; it allows the removal of surviving experimental units before the termination of the test. Refs. [9] and [10] provided a comprehensive reference on the subject of ⇑ Corresponding author. Tel.: +1 7572214628. E-mail address: [email protected] (T. Dey). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.07.025

S. Dey, T. Dey / Applied Mathematical Modelling 38 (2014) 974–982

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progressive censoring and its applications. For up to date references, the readers are referred to [11,12] and the references cited therein. Under the progressively Type-II censoring scheme, n units are placed on test at time zero and m failures are to be observed. When the first failure is observed, r 1 of surviving units are randomly selected and removed from the experiment. At the time of second failure, r 2 of the remaining n r 1 1 units are randomly selected and removed from the experiment. Finally, at the mth failure all the remaining surviving units rm ¼ n m r1 r2 r m1 are removed from the experiment. Note that, in this censoring scheme, r1 ; r2 ; . . . ; rm are all prefixed. [13] pointed out that, in a clinical test, the number of patients that dropped out at each stage is random and cannot be prefixed. In such cases, the pattern of removal at each failure is random. For some related classical estimation on progressive Type II censoring with binomial removals or random removals, one may refer to [14,15], [16–19] and the references cited therein. Recently, [20] discussed classical and Bayesian estimation problem on exponential distribution based on progressively Type-II censored data with random removals. [21] obtained both classical point and interval estimates of the unknown parameters of the Weibull model using Type II progressively censored samples under random removals. They also obtained the expected termination point under this censoring scheme. [22] studied Weibull distribution under progressive Type-II censoring for competing risk data with binomial removals. [23] studied generalized exponential distribution under progressive censoring with binomial removals. [24] studied Fréchet distribution under progressive Type II censoring with random removals. In this paper we consider one parameter Rayleigh distribution with the following probability density function (PDF), cumulative distribution function (CDF) respectively 2

f ðxjhÞ ¼ 2hxehx ;

x > 0; h > 0;

2

FðxjhÞ ¼ 1 ehx ;

ð1Þ

x > 0;

ð2Þ

here h is the scale parameter. 2. Estimation Let X ¼ ðX 1:m:n ; X 2:m:n ; . . . ; X m:m:n Þ be a progressively Type-II right censored sample from a life test of size m from a sample of size n, where lifetimes have a Rayleigh distribution with pdf as given by (1). With a pre-determined number of removal of units from the test, say R1 ¼ r 1 ; R2 ¼ r2 ; . . . ; Rm ¼ r m , the conditional likelihood function takes the form

m m L1 ðh; xjR ¼ rÞ ¼ C P f ðxi Þ½1 Fðxi Þri ¼ C hm P xi expðhTÞ; i¼1

ð3Þ

i¼1

P m 2 where T ¼ m i¼1 ðr i þ 1Þxi ; C ¼ nðn 1 r 1 Þðn 2 r 1 r 2 Þ ðn m þ 1 r 1 r m Þ and C ¼ 2 C. Presume that an individual unit being removed from the life test is independent of the others but with the same probability p. Then the number of units removed at each failure time follows a binomial distribution such that

PðR1 ¼ r 1 Þ ¼

n m r1 p ð1 pÞnmr1 ; r1

0 6 r1 6 n

ð4Þ

and

1 i X i1 X nm rk rk C Bn m C ri k¼1 ; ¼ r i1 ; . . . ; R1 ¼ r1 Þ ¼ B k¼1 A p ð1 pÞ @ ri 0

PðRi ¼ r i jRi1

ð5Þ

P where 0 6 r i 6 n m i1 k¼1 r k ; i ¼ 2; 3; . . . ; m 1. Also supposing further that Ri is independent of X i for all i. Therefore, the joint likelihood function X ¼ ðX 1 ; X 2 ; . . . ; X m Þ and R ¼ ðR1 ; R2 ; . . . ; Rm Þ can be expressed as

Lðh; p; x; rÞ ¼ Lðh; xjR ¼ rÞPðR ¼ rÞ;

ð6Þ

PðR ¼ rÞ ¼ PðRm1 ¼ r m1 jRm2 ¼ r m2 ; . . . ; R1 ¼ r 1 Þ PðR2 ¼ r 2 jR1 ¼ r 1 ÞPðR1 ¼ r 1 Þ:

ð7Þ

where

Substituting (4) and (5) into (7), we get m1 X

PðR ¼ rÞ ¼

ðn mÞ! m1 i¼1 r i !ðn

P

m

Pm1 i¼1

r i Þ!

ri

p i¼1 ð1 pÞðm1ÞðnmÞ

Pm1 i¼1

ðmiÞri

:

ð8Þ

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Now using (3), (6) and (8), we can write the full likelihood function as

Lðh; p; x; rÞ ¼ AL1 ðhÞL2 ðpÞ

ð9Þ

.

where A ¼ ½C ðn mÞ! ½ðn m h

Pm1 i¼1

ri Þ!

Qm1 i¼1

ri , does not depend on the parameter h and p and

m X

L1 ðhÞ ¼ hm e

ðri þ1Þx2i ;

ð10Þ

i¼1

m1 X

m1 X

ri

ðm1ÞðnmÞ

L2 ðpÞ ¼ p i¼1 ð1 pÞ

ðmiÞr i

i¼1

ð11Þ

:

3. Maximum likelihood estimation In this section, we discuss the process of obtaining the maximum likelihood estimates of the parameters h and p based on progressively Type-II censoring data with binomial removals. Both point and interval estimations of the parameters are derived. 3.1. Point estimation As it appears L1 does not involve p. Therefore, the MLE of hcan be derived by maximizing (10) directly. The log-likelihood function of L1 takes the following form (without constant term)

log L1 ðhjR ¼ rÞ ¼ m log h þ

m X

logðxi Þ hT

ð12Þ

i¼1

Thus, the MLE of his given by

^hMLE ¼ m : T

ð13Þ

Similarly, since L2 does not involve h, the maximum likelihood estimator of p can be derived by maximizing (11) directly. Thus the MLE of the binomial parameter p takes the following form

Pm1 ^MLE ¼ p

ri Pm1

i¼1

ðm 1Þðn mÞ

i¼1

ðm i 1Þri

ð14Þ

:

3.2. Interval estimation In this subsection, we derive the approximate confidence intervals of the unknown parameters a ¼ ðh; pÞ based on the asymptotic distributions of the MLE. The idea is to use the large sample approximation. It is known that the asymptotic dis^ is ða ^ aÞ ! N 2 ð0; I1 ðaÞÞ (see [25]), where I1 ðaÞ, the inverse of the observed information matrix of the tribution of the MLE a unknown parameters a ¼ ðh; pÞ can be written as

0 I ðaÞ ¼ @ 1

@

2

@

2

log Lðh;pÞ @h2 log Lðh;pÞ @p@h

11 A @ 2 log Lðh;pÞ @p2 @

2

log Lðh;pÞ @h@p

^MLE Þ ðh;pÞ¼ð^hMLE ;p

¼

v arð^hMLE Þ ^MLE ; ^hMLE Þ cov ðp

^MLE Þ cov ð^hMLE ; p

v arðp^MLE Þ

! :

The elements of I1 ðaÞ are as follows:

@ 2 log Lðh; pÞ

m ¼ 2; @h2 h @ 2 log Lðh; pÞ ¼ 0; @h@p Pm P 2 @ log Lðh; pÞ ðm 1Þðn mÞ m1 i¼1 r i i¼1 ðm iÞr i ¼ þ : 2 @p2 p2 ð1 pÞ

Using the above elements, oneq can deriveffi the approximate 100ð1 sÞ% confidence intervals of the parameters a ¼ ðh; pÞ in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^MLE zs=2 v arð^ ^MLE zs=2 v arðp ^MLE Þ, where zs=2 is the upper (s=2)th percentile of the stanhMLE Þ and p the following forms h dard normal distribution.

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4. Bayesian estimation This section discusses the Bayes procedure to derive the point and interval estimates of the parameter hand p based on progressively Type-II censored data with binomial removals. In our Bayesian analysis we have assumed only squared error loss function. It may be noted that if both parameters h and p are unknown, no conjugate prior exists. In such cases, there are several ways to choose the priors. One way is to consider the piecewise independent priors. In this article, the proposed priors for the parameters h; and p are considered as

g 1 ðhÞ / ha1 ebh ;

h; a; b > 0;

g 2 ðpÞ / pc1 ð1 pÞd1 ;

ð15Þ

0 < p < 1; c; d > 0;

respectively. Thus the joint prior distribution for h and p is given by

gðh; pÞ / ha1 ebh pc1 ð1 pÞd1 :

ð16Þ

The joint posterior distribution of h and pis obtained after simplification as

hf1 ehb pc1 ð1 pÞd1

pðh; pjx; rÞ ¼ R 1 R 1 0

0

hf1 ehb pc1 ð1 pÞd1 dpdh

ð17Þ

;

P P where f ¼ a þ m; b ¼ b þ m ðri þ 1Þx2i ; c ¼ c þ m1 i¼1 i¼1 r i , Pm1 d ¼ ðm 1Þðn mÞ þ d i¼1 ðm iÞr i . Therefore the marginal posterior pdfs of hand p are respectively

bf hf1 ebh Bðc; dÞCðfÞ

p1 ðhjx; rÞ ¼

Z

1

pc1 ð1 pÞd1 dp ¼

0

bf

CðfÞ

hf1 ebh

ð18Þ

and

bf pc1 ð1 pÞd1 Bðc; dÞCðfÞ

p2 ðpjx; rÞ ¼

Z

1

hf1 ebh dh ¼

0

1 pc1 ð1 pÞd1 : Bðc; dÞ

ð19Þ

^B along with their associated posterior risk, say Under squared error loss function, the Bayes estimators of hand p, say ^ hB and p ^B Þ are given as follows Rð^ hB Þ and Rðp

^hB ¼

Z

1

0

Z

Rð^hB Þ ¼ ^B ¼ p

Z

f hp1 ðhjx; rÞdh ¼ ; b 1

2 f h2 p1 ðhjx; rÞdh ½^hB ¼ 2 ; b

0 1

0

^B Þ ¼ Rðp

pp2 ðpjx; rÞdp ¼ Z

c ; cþd

1

cd

2

^B ¼ p2 p2 ðpjx; rÞdp ½p

0

ðc þ dÞ2 ðc þ d þ 1Þ

:

Similarly under squared error loss function, we obtain the posterior risk of the maximum likelihood estimators of h and p, say ^MLE Þ are specified as follows Rð^ hMLE Þ and Rðp

Rð^hMLE Þ ¼

Z

1

0

^MLE Þ ¼ Rðp

Z

1

2 fðf þ 1Þ m2 h2 p1 ðhjx; rÞdh ½^hMLE ¼ ; T b2

0

P

m1 cðc þ 1Þ i¼1 r i ^MLE ¼ p p2 ðpjx; rÞdp ½p P ðc þ dÞðc þ d þ 1Þ ðm 1Þðn mÞ m1 i¼1 ðm i 1Þr i 2

2

!2 :

4.1. Two-sided Bayes probability interval and credible interval The Bayesian method of interval estimation is more straightforward than the classical method of confidence intervals. Once the marginal posterior distribution of hhas been obtained, a symmetric 100ð1 sÞ% two-sided Bayes probability interval (TBPI) of h, denoted by ½hL ; hU , can be obtained by solving the following two equations (see [26, page 208–209]).

Z 0

hL

s p1 ðhjx; rÞdh ¼ ; 2

Z

1 hU

s p1 ðhjx; rÞdh ¼ ; 2

for the limits hL and hU . Similarly, a symmetric 100ð1 sÞ% TBPI of p, denoted by

978


Z

pL

s p2 ðpjx; rÞdp ¼ ;

Z

2

0

1

s p2 ðpjx; rÞdp ¼ ; 2

pU

for the limits pL and pU . Now we compute the highest posterior density (HPD) credible intervals for h and p. Since p1 ðhjx; rÞ is unimodal, the corresponding 100ð1 cÞ% HPD credible interval ½HhL ; HhU can be obtained from the simultaneous solution of the following equations

PðHhL < h < HhU ¼

Z

HhU

HhL

p1 ðhjx; rÞdh ¼ 1 cp1 ðHhL jx; rÞ ¼ p1 ðHhU jx; rÞ:

Similarly for the parameter p, the 100ð1 cÞ% HPD credible interval ½HpL ; HpU can be obtained from the simultaneous solution of the following equations

PðHpL < p < HpU ¼

Z

HpU p

HL

p2 ðpjx; rÞdp ¼ 1 cp2 ðHpL jx; rÞ ¼ p2 ðHpU jx; rÞ:

5. Numerical experiments and data analysis 5.1. Numerical experiments In this section we present several experimental results to illustrate the behavior of the proposed method for different sample sizes, different effective sample sizes, and for different priors using progressively Type-II censoring under binomial removal scheme. We have considered different sample sizes; n ¼ 20; 25; 30, different effective sample sizes; m ¼ 10; 15 with a ¼ 2; b ¼ 1; c ¼ d ¼ 2. In all cases we have taken, without loss of generality, h ¼ 1 and p ¼ 0:5. We have generated samples for a given n and m along with a sampling scheme by using binomial removal technique. We computed the MLEs of the unknown parameters using the method proposed in Section 3, as well as the Bayes estimates based on the method proposed in Section 4. The performance of the estimates are compared based on the average bias and the corresponding mean squared error (MSE) of the estimates under 10,000 replications. In addition we also computed the 95% confidence intervals (CIs), the 95% two-sided Bayesian probability intervals (TBPIs) and the HPD credible intervals based on the same 10,000 replications. Simulation study results are summarized in Tables 1 and 2. Tables show that as sample size increases, the MSEs decrease. The performance of the Bayes estimates are better than the MLEs, in terms of both the biases and the MSEs. Both 95% CIs and TBPIs shows good coverage of the true value of the parameters being considered. The TBPIs are in general, of shorter length than the CIs, and the HPDs are of shorter length than the TBPIs. 6. Data analysis Here we consider n ¼ 23 deep-groove ball bearing failure times. The 23 failure times are

0:1788 0:2892 0:33

0:4152 0:4212

0:4560 0:4848

0:5184

0:5196 0:5412 0:5556 0:6780 0:6864

0:6864 0:6888

0:8412

0:9312 0:9864 1:0512 1:0584 1:2792 1:2804 1:7340: This data set originally used by [27], is also discussed in [28]. The observations are the number of million revolutions before failure for each 23 ball bearings ordered according to life endurance. [29] argued that the Rayleigh distribution provides a reasonable fit to the ball bearing data. Here we analyze this data set from the perspective of progressive Type-II censoring using binomial removal scheme. We are interested to analyze this data set based on effective sample size m ¼ 15. We have used several values of p to generate several removal schemes; our analysis is performed on the data generated from the

Table 1 The average bias (AB), the mean squared error (MSE) for the estimates of h for different sample sizes and the different effective sample sizes. n

20 20 25 25 30 30

m

10 15 10 15 10 15

MLE

Bayes

AB

MSE

CI

Cover (%)

AB

MSE

TBPI

Cover

HPD

Cover (%)

0.103 0.086 0.085 0.064 0.110 0.071

0.164 0.098 0.151 0.084 0.173 0.084

(0.419, 1.787) (0.537, 1.637) (0.412, 1.757) (0.525, 1.603) (0.422, 1.798) (0.529, 1.618)

95.5 92.2 95.3 95.9 96.2 95.8

0.068 0.054 0.044 0.041 0.056 0.035

0.093 0.075 0.065 0.056 0.074 0.067

(0.609, 1.934) (0.667, 1.745) (0.601, 1.905) (0.653, 1.714) (0.613, 1.945) (0.657, 1.724)

94.7 94.7 95.9 95.2 95.1 95.8

(0.617, 1.914) (0.671, 1.729) (0.629, 1.902) (0.664, 1.675) (0.638, 1.915) (0.672, 1.698)

94.5 94.4 95.7 95.1 95.1 95.7%

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S. Dey, T. Dey / Applied Mathematical Modelling 38 (2014) 974–982 Table 2 The average bias (AB), the mean squared error (MSE) for the estimates of p for different sample sizes and the different effective sample sizes. n

20 20 25 25 30 30

m

10 15 10 15 10 15

MLE

Bayes

AB

MSE

CI

Cover ( (%))

AB

MSE

TBPI

Cover

HPD

Cover ( (%))

0.026 0.046 0.016 0.032 0.010 0.016

0.015 0.029 0.009 0.015 0.006 0.008

(0.309, 0.749) (0.233, 0.833) (0.339, 0.694) (0.315, 0.749) (0.356, 0.664) (0.338, 0.693)

93.2 87.5 93.5 92.8 94.5 94.4

0.018 0.022 0.012 0.022 0.008 0.011

0.009 0.012 0.006 0.009 0.005 0.006

(0.323, 0.707) (0.271, 0.761) (0.347, 0.674) (0.326, 0.713) (0.363, 0.652) (0.347, 0.674)

96.4 96.7 95.8 96.7 95.7 96.4

(0.331, 0.698) (0.289, 0.725) (0.359, 0.651) (0.335, 0.689) (0.381, 0.642) (0.361, 0.672)

96.3 96.6 95.8 96.5 95.6 96.3

original ball bearing data set. Table 4 summarizes the goodness-of-fit test results from Kolmogorov–Smirnov (KS) test and the Chi-square test for four different sampling schemes. The results indicate that the Rayleigh distribution provides quite reasonable fit for all four sampling schemes. Fig. 1 is the fitted distribution and empirical distribution plot under four censoring schemes based on the ball bearing data. The plots depict that Rayleigh model gives a good fit to the data set as both the fitted distribution and empirical distribution are quite close to each other. For Bayes estimation, we have used non-informative priors for both h and p ða ¼ b ¼ c ¼ d ¼ 0Þ (see table 3). Table 5 represents the output of the ball bearing data analysis. The performance of the estimates are compared with respect to their corresponding risk performances. The Bayes estimates perform better than the MLEs for h. As p increases in the removal scheme, the risk estimate also increases. The HPDs are of the shortest length than that of the CIs and the TBPIs. 7. Expected experimentation time In practical applications, it is often useful to have an idea of the duration of a lifetime test as the time required to complete the test is directly related to cost. Hence it is imperative to compute the expected time required to complete a life test. Under a Type II progressive censoring scheme with binomial removals, the expected termination time for the experiment is given by the expectation of the mth order statistic X ðmÞ . The conditional expectation of X ðmÞ , for a fixed set of R ¼ ðR1 ¼ r1 ; . . . ; Rm1 ¼ rm1 Þ is (see [9])

E½X ðmÞ jR ¼ r ¼ CðrÞ

r1 X l1 ¼0

rm X lm ¼0

ð1ÞA

r1 l1

rm lm

m1 i¼1 hðli Þ

P

Z

1

xf ðxÞF hðli Þ1 ðxÞdx;

ð20Þ

0

P Pm1 where A ¼ m i¼1 li ; CðrÞ ¼ nðn r 1 1Þðn r 1 r 2 2Þ n i¼1 ðr i þ 1Þ, hðli Þ ¼ l1 þ l2 þ þ li þ i and i is the number of live units removed from experimentation (or the number of failure units). Let,

Ball bearing data scheme 2

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.0

1.0


0.5

1.0

1.5

0.5

1.5


0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1.0

1.0


1.0

0.5

1.0

1.5

0.5

1.0

1.5

Fig. 1. The empirical and fitted cumulative distribution function plot for ball bearing data under four different sampling schemes.

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S. Dey, T. Dey / Applied Mathematical Modelling 38 (2014) 974–982 Table 3 Several censoring schemes. p

Number

Scheme

0.2 0.5 0.7 0.9

[1] [2] [3] [4]

(3 (4 (7 (5

3 2 0 2

0 2 1 1

1 0 0 0

0 0 0 0

1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0) 0) 0) 0)

Table 4 The goodness-of-fit test results for ball bearing data based on four sampling schemes. Scheme

KS statistic

p-Value

Chi-square statistic

p-Value

[1] [2] [3] [4]

0.1206 0.1790 0.1752 0.1768

0.9623 0.6579 0.6834 0.6728

0.0532 0.0453 0.0417 0.0432

0.8174 0.8313 0.8381 0.8352

Table 5 Data analysis result of the Ball bearing data. Scheme

^MLE Þ Rðp

CIp

^B Þ Rðp

TBPIp

HPDp

[1] [2] [3] [4]

0.021 -0.004 -0.016 -0.040

(0.199, 0.643) (0.312, 0.830) (0.352, 0.914) (0.399, 0.933)

0.010 0.012 0.013 0.013

(0.164, 0.633) (0.328, 0.764) (0.458, 0.908) (0.480, 0.973)

(0.168, 0.621) (0.334, 0.729) (0.469, 0.884) (0.482, 0.958)

Rð^ hMLE Þ

CIh

Rð^ hB Þ

TBPIh

HPDh

0.201 0.203 0.203 0.204

(0.506, 1.544) (0.510, 1.555) (0.517, 1.608) (0.524, 1.667)

0.069 0.070 0.075 0.075

(0.631, 1.662) (0.641, 1.694) (0.668, 1.726) (0.678, 1.786)

(0.635, 1.653) (0.644, 1.675) (0.672, 1.685) (0.689, 1.727)

[1] [2] [3] [4]

S¼

Z

1

xf ðxÞF hðli Þ1 ðxÞdx ¼

0

Z 0

1

hðl i Þ1 X Cð32Þ hðli Þ 1 2 2 hðli Þ1 2hx2 ehx 1 ehx dx ¼ ð1Þk : 3 1 k h2 ðk þ 1Þ2 k¼0

Plugging this value into the right hand side of (20), the expected experimentation time is given by

r1 r1 rm i Þ1 rlmm hðlX X X Cð32Þ hðli Þ 1 A l1 E½X ðmÞ jR ¼ r ¼ CðrÞ ð1Þ Qm1 ð1Þk : 3 1 k h2 ðk þ 1Þ2 k¼0 l1 ¼0 lm ¼0 i¼1 hðli Þ

ð21Þ

The expected termination time of Type-II progressive sample without removal is defined as the expected value of the m-th failure time, denoted by X ðmÞ

m1 3 n m1 X Cð2Þ E½X ðmÞ ¼ m ð1Þk 1 k 3 m k¼0 h2 ðk þ 1Þ2

ð22Þ

Eq. (22) can be derived from (20) by setting ri ¼ 0 for all i ¼ 1; 2; . . . ; m 1. Similarly, the expected termination point time of the complete sample can be derived from (22) by setting m ¼ n as

E½X ðnÞ ¼ n

n1 X ð1Þk k¼0

n1 k

Cð32Þ 3

1 2

h ðk þ 1Þ2

ð23Þ

For the Type-II progressive censoring with random removals, the expected termination point is given by E½X ðmÞ ¼ ER ½EðX ðmÞ jRÞ. For binomial removals, we have

E½X ðmÞ ¼

gðr 1 Þgðr 2Þ X X r 1 ¼0r 2 ¼0

gðr m1 Þ X

PðR ¼ rÞE½X ðmÞ jR ¼ r

ð24Þ

r m1 ¼0

where gðr1 Þ ¼ n m; gðri Þ ¼ n m r1 r2 r i1 ; i ¼ 2; 3; . . . ; m 1, and PðR ¼ rÞ is given by (8). To compare (23) and (24), we compute the ratio of expected experiment time (REET) under the Type-II progressive censoring with binomial removals, over the expected termination point for complete sample which is given by

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Fig. 2. The REET plot for Type-II progressively censored data under binomial removal scheme for different values of h and m with varying values of p.

REET ¼

E½X ðmÞ E½X ðnÞ

ð25Þ

The ratio REET is valuable in determining the shortest experiment time if the sample size n is large. When REET is closer to 1, the termination point will be closer to the complete sample. We can also study the influence of the binomial probability removals p on the expected termination point by analyzing REET for various ps. 7.1. Numerical experiment We perform an experiment to study the impact of n; m; p and h in Type-II progressive sampling on REET as described in (25). To do so we have used m ¼ 3; 9; p ¼ 0:05; 0:3; 0:6; 0:9 and h ¼ 0:5; 1. For each condition, we computed REET varying the

Fig. 3. The REET plot for Type-II progressively censored data under binomial removal scheme for different values of n with varying values of p.

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sample size n. Fig. 1 depicts the experiment for four different situations. For large values of p and m, we see that the REET approaches 1 quite sharply. It is notable that the experimentation time is closely dependent on the removal probability p. From Fig. 2, it is observed that the expected termination time for Type-II progressive censoring sample is getting close to the complete sample when m is increasing. But for fixed n and m, the values of REET and the expected termination time of the progressive censoring with binomial removals are increasing as p increases. Hence the simulation study confirms that the experimentation time under progressively Type-II censored scheme with binomial removals is influenced by the removal probability p. If p is not too large, say less than 0.5, the reduction has no impact. In contrast, if p is large, even when the number of test units n is large, most units would be removed early on in the process (see Fig. 3). 8. 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