# Bayesian Estimation Based on Rayleigh Progressive Type II Censored Data with Binomial Removals

Authors: Reza Azimi; Farhad Yaghmaei
Journal: Journal of Quality and Reliability Engineering
 (2013)

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## Abstract

This study considers the estimation problem for the parameter and reliability function of Rayleigh distribution under progressive type II censoring with random removals, where
the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood and Bayesian procedures to obtain the estimators of parameter and reliability function of Rayleigh distribution. We also construct the confidence intervals for the parameter of Rayleigh distribution. Monte Carlo simulation method is used to generate a progressive type II censored data with binomial removals from Rayleigh distribution, and then these data are used to compute the point and interval estimations of the parameter and compare both the methods used with different random schemes.


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## Body

## 1. Introduction

The Rayleigh distribution provides a population model which is useful in several areas of statistics. In the literature, many researcher studied properties of the Rayleigh distribution, particularly in life testing and reliability. Life testing experiments often deal with censored sample in order to estimate the parameters involved in the life distribution. The cumulative distribution function (CDF), probability density function (PDF), and the reliability function of the Rayleigh distribution with parameterσ>0 are
(1)F(x∣σ)=1-exp{-x22σ2},x>0,(2)f(x∣σ)=xσ2exp{-x22σ2},(3)R(x∣σ)=exp{-x22σ2},
respectively. Inferences for σ in the Rayleigh distribution have been discussed by several authors. Harter and Moore [1] derived an explicit form for the maximum likelihood estimator (MLE) of σ based on type II censored data. Moreover, Bayesian estimation and prediction problems for the σ based on doubly censored sample have been considered by Fernández [2] and Raqab and Madi [3]. Wu et al. [4] have derived the Bayesian estimator and prediction intervals based on progressively type II censored samples. A recent account on progressive censoring schemes can be obtained in the monograph by Balakrishnan and Aggarwala [5] or in the excellent review article by Balakrishnan [6].Suppose thatn units are placed on a life test and the experimenter decides beforehand quantity m, the number of units, to be failed. Now at the time of the first failure, R1 of the remaining n-1 surviving units are randomly removed from the experiment. Continuing, at the time of the second failure, R2 of the remaining n-R1-2 units are randomly drawn from the experiment. Finally, at the time of the mth failure, all the remaining Rm=n-m-R1-R2-⋯-Rm-1 surviving units are removed from the experiment. Note that, in this scheme, R1,R2,…,Rm are all prefixed. However, in some practical situations, these numbers may occur at random. For example, in some reliability experiments, an experimenter may decide that it is inappropriate or too dangerous to carry on the testing on some of the tested units even though these units have not failed. In such cases, the pattern of removal at each failure is random [7, 8]. This leads to progressive censoring with random removals (illustrated by Table 1). There have been several references about the statistical inference on lifetime distributions under progressive censoring with random removals. Wu et al. [9] Wu and Chang [10] considered the estimation problem for the Burr type-XII distribution and Pareto distribution based on progressively censored samples with random removals, where the number of units removed at each failure time has a discrete uniform removal pattern and binomial distribution.Table 1
A schematic representation of the progressive type II censoring with binomial removals.
Process
The number in life testing
Failures
Binomial removals
Remains
1
n
1
R
1
~
B
(
n
-
m
,
p
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1
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2
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2
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## 2. Maximum Likelihood Estimation

LetX1<X2<⋯<Xm be the ordered failure times out of n randomly selected times, where m is predetermined before the test. At the ith failure, Ri items are removed from the test. For progressive censoring with predetermined number of removals R=(R1=r1,…,Rm-1=r1), the likelihood function can be defined as the following form:
(4)L(σ,x∣R)=c(r)∏i=1mf(xi∣σ)[1-F(xi∣σ)]ri,
where c(r)=n(n-1-r1)⋯(n-∑i=1m-1(ri+1)). Equation (4) is derived conditional on ri. Each ri can be of any integer value between 0 and n-m-∑i=1i-1(rj). It is different from progressive censoring with fixed removal that ri is a random number and is assumed to follow a binomial distribution with parameter p. It means that each unit leaves with equal probability p and the probability of ri units leaving after the ith failure occurs is
(5)P(R1=r1)=(n-mr1)pr1(1-p)n-m-r1,P(Ri=ri∣Ri-1=ri-1,…,R1=r1)=(n-m-∑j=1i-1rjri)pri(1-p)n-m-∑j=1m-1rj,
where 0≤ri≤n-m-∑j=1i-1rj(i=1,…,m-1). Furthermore, we assume that Ri is independent of Xi for all i. The joint likelihood function of X=(X1,X2,…,Xm) and R=(R1,R2,…,Rm) can be found as
(6)L(σ,x,p)=L(σ,x∣R)P(R,p),
where P(R,p) is the joint probability distribution of R=(r1,r2,…,rm) and in particular
(7)P(R,p)=P(Rm=rm∣Rm-1=rm-1,…,R1=r1)×P(R2=r2∣R1=r1)P(R1=r1).
Therefore
(8)P(R,p)=(n-m)!(n-m-∑j=1m-1rj)!∏j=1m-1rjp∑j=1m-1rj×(1-p)(m-1)(n-m)-∑j=1m-1(m-j)rj.
Substituting (1) and (2) into (6), the likelihood function takes the following form:
(9)L(σ,x,p)∝σ-2mexp{-12σ2∑i=1m(Ri+1)xi2}p∑j=1m-1rj×(1-p)(m-1)(n-m)-∑j=1m-1(m-j)rj.
Thus, the log-likelihood function takes the following form:
(10)l=lnL=-2mlnσ-12σ2∑i=1m(Ri+1)xi2+(∑j=1m-1rj)lnP+((m-1)(n-m)-∑j=1m-1(m-j)rj)ln(1-P).
The first partial derivatives of log-likelihood function l with respect to σ and p are
(11)∂l∂σ=-2mσ+∑i=1m(ri+1)xi2σ3=0,∂l∂P=∑j=1m-1rjP-(m-1)(n-m)-∑j=1m-1(m-j)rj1-P=0.
Solving these equations in σ and p, we get the MLE of σ and p as in the following forms:
(12)σ^=∑i=1m(ri+1)xi22m,p^=∑j=1m-1rj(m-1)(n-m)-∑j=1m-1(m-j)rj+∑j=1m-1rj.
It is clear that the MLE of reliability function, R(x0), with fixed x0>0, can be obtained by
(13)R^=exp{-x022σ^2}.


## 3. Confidence Intervals forσ

LetYi=Xi2/2σ2,i=1,…,m. It is easy to show that Y1,…,Ym is a progressively type II censored sample from the standard exponential distribution. For a fixed set of R=(r1,…,rm), let us consider the following transformation:
(14)Z1=nY1,Z2=(n-r1-1)(Y2-Y1),⋮Zm=(n-r1-⋯-rm-m+1)(Ym-Ym-1).Thomas and Wilson [11] showed that the generalized spacings Z1,…,Zm, as defined in (14), are all independent and identically distributed as standard exponential. Hence, 2Z1 has a Chi-square distribution with 2 degrees of freedom. We can also write the numerator of σ^2 as the sum of m-independent generalized spacings. That is, 2∑i=1mZi=∑i=1m((ri+1)Xi2/σ2). Therefore, we can find that, conditionally on a fixed set of R=(r1,…,rm),2mσ^2/σ2 has a Chi-square distribution with 2m degrees of freedom.Confidence interval forσ2 can be obtained through the pivotal quantity 2mσ^2/σ2~χ(2m)2 and can be the percentile of Chi-square distribution with 2m degrees of freedom and right-tail probability α. Then
(15)(2mσ^2χ1-(α/2)(2m)2,2mσ^2χ(α/2)(2m)2)
since σ2 is increasing function of σ. Then confidence interval for σ is
(16)(2mσ^2χ1-(α/2)(2m)2,2mσ^2χ(α/2)(2m)2).


## 4. Bayesian Estimation

In this section, we use the Bayes procedure to derive the point estimates of the parametersσ and p and reliability function R(x0) based on progressively type II censoring data with binomial removals. For this purpose we assume the parameters σ and p behave as independent random variables. In this paper, for parameter σ we consider prior distribution of the form
(17)π1(σ)=abΓ(b)2b-1σ-2b-1exp{-a2σ2},
where a>0 and b>0. This density is known as the square-root inverted-gamma distribution (see [12, page 119]) and, independently from parameter σ, p has a beta prior distribution with parameters α and β and PDF
(18)π2(p)=1B(α,β)pα-1(1-p)β-1,0<p<1;α,β>0.
Based on the prior π1(σ) and π2(p), the joint prior PDF of (σ,p) is
(19)π(σ,p)=π1(σ)π2(p),σ>0,0<p<1(20)π(σ,p)=abB(α,β)Γ(b)2b-1pα-1π(σ,p)×(1-p)β-1σ-2b-1exp{-a2σ2}.
It follows, from (9) and (20), that the joint posterior distribution of (σ,p) is given by
(21)π(σ,p∣x,r)=a*b*B(α*,β*)Γ(b*)2b*-1pα*-1×(1-p)β*-1σ-2b*-1exp{-a*2σ2},
where a*=a+∑i=1m(Ri+1)xi2,b*=m+b,α*=α+∑j=1m-1rj,β*=β+(m-1)(n-m)-∑j=1m-1(m-j)rj.Therefore, the marginal posterior PDFs ofσ and p are given, respectively, by
(22)π(σ∣x,r)=a*b*Γ(b*)2b*-1σ-2b*-1exp{-a*2σ2},a*>0,b*>0,(23)π(p∣x,r)=1B(α*,β*)pα*-1(1-p)β*-1,0<p<1;α*,β*>0
Substituting σ2=-x02/2logs into (22), we obtain the posterior probability density function of s=R(x0∣σ) as
(24)π(s∣x,r)=1Γ(b*)(a*x02)b*(-logs)b*-1sa*/x02-1
for 0<s<1.


### 4.1. Bayesian Estimation under Squared Error Loss Function

Under squared error loss, the Bayes estimator ofσ is the mean of posterior density (22), which is given by
(25)σ^S=a*2Γ(b*-1/2)Γ(b*).
And for parameter p, the Bayes estimator of p is
(26)p^S=α*α*+β*.
Finally, the Bayes estimators of R(x0) are found to be
(27)R^S=(a*a*+x02)b*.


### 4.2. Bayesian Estimation under Entropy Loss Function

In many practical situations, it appears to be more realistic to express the loss in terms of the ratioθ^/θ,θ=(σ,p). In this case, Calabria and Pulcini [13] point out that a useful asymmetric loss function is the entropy loss function:
(28)L(δ)∝[δd-dloge(δ)-1],
where
(29)δ=θ^θ,
whose minimum occurs at θ^=θ. The latter has been used in Dey et al., [14] and Dey and Liu [15], in the original form having d=1:
(30)L(δ)=b[δ-loge(δ)-1],b>0.
Therefore, the Bayesian estimators of parameters θ=(σ,p) under the entropy loss function with d=1, denoted by θ^E, is given by the following equation:
(31)θ^E=[E(1θ∣x,r)]-1.
Therefore we have
(32)σ^E=a*2Γ(b*)Γ(b*+1/2),(33)p^E=α*-1α*+β*-1,(34)R^E=(1-x02a*)b*.


### 4.3. Bayesian Estimation under Precautionary Loss Function

Precautionary loss function which is asymmetric, was introduced by Norström [16]:
(35)L(θ^,θ)=(θ^-θ)2θ^.
The Bayes estimator under the latter loss function is denoted by θ^P and may be obtained by solving the following equation:
(36)θ^P2=E(θ2∣x,r).
The Bayesian estimators of the parameters σ, p and R(t) under asymmetric precautionary loss function are given, respectively, by
(37)σ^P=a*2(b*-1),(38)p^P=(α*(α*+1)(α*+β*+1)(α*+β*))1/2,(39)R^P=(a*a*+2x02)b*/2.


## 4.1. Bayesian Estimation under Squared Error Loss Function

Under squared error loss, the Bayes estimator ofσ is the mean of posterior density (22), which is given by
(25)σ^S=a*2Γ(b*-1/2)Γ(b*).
And for parameter p, the Bayes estimator of p is
(26)p^S=α*α*+β*.
Finally, the Bayes estimators of R(x0) are found to be
(27)R^S=(a*a*+x02)b*.


## 4.2. Bayesian Estimation under Entropy Loss Function

In many practical situations, it appears to be more realistic to express the loss in terms of the ratioθ^/θ,θ=(σ,p). In this case, Calabria and Pulcini [13] point out that a useful asymmetric loss function is the entropy loss function:
(28)L(δ)∝[δd-dloge(δ)-1],
where
(29)δ=θ^θ,
whose minimum occurs at θ^=θ. The latter has been used in Dey et al., [14] and Dey and Liu [15], in the original form having d=1:
(30)L(δ)=b[δ-loge(δ)-1],b>0.
Therefore, the Bayesian estimators of parameters θ=(σ,p) under the entropy loss function with d=1, denoted by θ^E, is given by the following equation:
(31)θ^E=[E(1θ∣x,r)]-1.
Therefore we have
(32)σ^E=a*2Γ(b*)Γ(b*+1/2),(33)p^E=α*-1α*+β*-1,(34)R^E=(1-x02a*)b*.


## 4.3. Bayesian Estimation under Precautionary Loss Function

Precautionary loss function which is asymmetric, was introduced by Norström [16]:
(35)L(θ^,θ)=(θ^-θ)2θ^.
The Bayes estimator under the latter loss function is denoted by θ^P and may be obtained by solving the following equation:
(36)θ^P2=E(θ2∣x,r).
The Bayesian estimators of the parameters σ, p and R(t) under asymmetric precautionary loss function are given, respectively, by
(37)σ^P=a*2(b*-1),(38)p^P=(α*(α*+1)(α*+β*+1)(α*+β*))1/2,(39)R^P=(a*a*+2x02)b*/2.


## 5. Numerical Study

In our study, we firstly generate the numbers of progressive censoring with binomial removalsri(i=1,2,…,m), and then we get the progressive censoring with binomial removals samples from Rayleigh distribution by the Monte-Carlo method. The steps are as follows.Step 1.
Generate a group values(40)ri~B(n-m-∑j=1i-1rj,p),rm=n-m-∑i=1m-1ri,i=1,2,…,m-1
according to the relevant value of p.Step 2.
Simulatem-independent exponential random variables Z1,Z2,…,Zm.
This can be done using inverse transformationZi=-ln(1-Ui), where Ui are independent uniform (0,1) random variables.Step 3.
Set(41)Xi=Z1n+Z2n-R1-1+Z3n-R1-R2-2+⋯+Zin-R1-R2-⋯-Ri-1-i+1
for i=1,2,…,m. This is the required progressively type II censored sample with binomial removals from the standard exponential distribution.Step 4.
Finally, we setYi=F-1(1-exp(-Xi)), for i=1,2,…,m, where F-1(·) is the inverse cumulative distribution function of the Rayleigh distribution. Then Y1,Y2,…,Ym is the required progressively type II censored sample with binomial removals from the Rayleigh distribution.Step 5.
We compute the MLE of parametersσ and p by (12), for reliability function by (13) and 95% confidence intervals by (16).Step 6.
We compute the the Bayes estimates of parameters and reliability function, respectively, using (25), (32), (37), (26), (33), (38), (27), (34), and (39).Step 7.
We repeat the above steps 2000 times. We then obtain the means and the MSEs (mean-squared error) for different censoring sizesm and censoring schemes where
(42)MSE=2000-1∑i=12000(ϕ-ϕ^i)2,
and ϕ^ is the estimator of ϕ.In all above cases the prior parameters chosen as(a=2,  b=1) and (α=3,β=2) which yield the generated value of σ=1 as the true value. The true values of R(x0) in x0=0.5 is obtained R(0.5)=0.8824969. The results are summarized in Tables 2 and 3.Table 2
Averaged values of MSEs for estimates of the parametersθ and p.
n
m
σ
^ MSE
σ
^
S MSE
σ
^
E MSE
σ
^
P MSE
CI
p
^ MSE
p
^
S MSE
p
^
E MSE
p
^
P MSE
P = 0.1
10
5
0.9727
1.0484
1.0749
1.0019
(0.6796, 1.7070)
0.1333
0.2500
0.2105
0.1336
0.0514
0.0416
0.0469
0.0359
0.0011
0.0225
0.0122
0.0011
7
0.9794
1.0334
1.0520
0.9996
(0.7170, 1.5447)
0.0769
0.2222
0.1764
0.1139
0.0343
0.0292
0.0319
0.0263
0.0005
0.0149
0.0058
0.0001
20
10
0.9873
1.0250
1.0379
1.0009
(0.7554, 1.4258)
0.1228
0.1612
0.1475
0.0673
0.0251
0.0226
0.0239
0.0209
0.0005
0.0037
0.0022
0.0010
15
0.9889
1.0141
1.0226
0.9979
(0.7902, 1.3218)
0.0975
0.1521
0.1333
0.0627
0.0165
0.0153
0.0158
0.0146
5 × 10−6
0.0027
0.0011
0.0013
P = 0.4
10
5
0.9700
1.0461
1.0726
0.9997
(0.6778, 1.7024)
0.5000
0.5333
0.5000
0.4000
0.0522
0.0419
0.0471
0.0363
0.01000
0.0177
0.0100
0.0000
7
0.9805
1.0345
1.0532
1.0006
(0.7178, 1.5463)
0.2500
0.3846
0.3333
0.2517
0.0371
0.0318
0.0346
0.0287
0.0225
0.0002
0.0044
0.0219
20
10
0.9842
1.0221
1.0349
0.9980
(0.7530, 1.4213)
0.3703
0.4062
0.3870
0.2646
0.0243
0.0216
0.0229
0.0202
0.0008
0.0009
0.0001
0.0183
15
0.9897
1.0149
1.0234
0.9986
(0.7909, 1.3230)
0.4545
0.5000
0.4666
0.3638
0.0163
0.0151
0.0157
0.0144
0.0029
0.0100
0.0044
0.0013
P = 0.6
10
5
0.9706
1.0466
1.0730
1.0001
(0.6782, 1.7034)
0.7142
0.6666
0.6363
0.5547
0.0511
0.0411
0.0462
0.03557
0.0130
0.0044
0.0013
0.0020
7
0.9818
1.0356
1.0542
1.0017
(0.7188, 1.5485)
0.2727
0.3750
0.3333
0.2406
0.0343
0.0296
0.0323
0.0265
0.1071
0.0506
0.0711
0.1291
20
10
0.9906
1.0280
1.0409
1.0038
(0.7579, 1.4305)
0.6666
0.6500
0.6315
0.5307
0.0240
0.0219
0.0233
0.0201
0.0044
0.0025
0.0009
0.0047
15
0.9913
1.0164
1.0249
1.0001
(0.7922, 1.3251)
0.4545
0.5000
0.4666
0.3638
0.0161
0.0150
0.0156
0.0143
0.0211
0.0100
0.0177
0.0557
P = 0.9
10
5
0.9746
1.0500
1.0766
1.0034
(0.6810, 1.7104)
0.7142
0.6666
0.6363
0.5547
0.0499
0.0408
0.0461
0.0349
0.0344
0.0544
0.0695
0.1192
7
0.9779
1.0322
1.0508
0.9984
(0.7159, 1.5423)
0.7500
0.6666
0.6250
0.5577
0.0370
0.0315
0.0341
0.0285
0.0225
0.05447
0.0756
0.1171
20
10
0.9882
1.0258
1.0387
1.0017
(0.7560, 1.4270)
0.9090
0.8125
0.8000
0.7373
0.0248
0.0223
0.0237
0.0207
8 × 10−5
0.0076
0.0100
0.0264
15
0.9931
1.0181
1.0266
1.0018
(0.7936, 1.3274)
0.6250
0.6153
0.5833
0.4934
0.0153
0.0143
0.0149
0.0136
0.0756
0.0810
0.1002
0.1653Table 3
Averaged values of MSEs for estimates of the reliability function.
n
m
R
^ MSE
R
^
S MSE
R
^
E MSE
R
^
P MSE
P = 0.1
10
5
0.8578
0.8353
0.8297
0.8379
0.0052
0.0078
0.0094
0.0072
7
0.8662
0.8505
0.8472
0.8520
0.0030
0.0042
0.0048
0.0040
20
10
0.8714
0.8606
0.8586
0.8615
0.0018
0.0024
0.0026
0.0023
15
0.8764
0.8694
0.8682
0.8699
0.0010
0.0012
0.0013
0.0012
P = 0.4
10
5
0.8589
0.8366
0.8310
0.8392
0.0053
0.0079
0.0094
0.0072
7
0.8660
0.8503
0.8470
0.8519
0.0030
0.0042
0.0048
0.0040
20
10
0.8690
0.8581
0.8560
0.8590
0.0020
0.0026
0.0029
0.0025
15
0.8744
0.8673
0.8661
0.8679
0.0010
0.0013
0.0014
0.0012
P = 0.6
10
5
0.8592
0.8369
0.8313
0.8394
0.0052
0.0078
0.0092
0.0071
7
0.8672
0.8516
0.8484
0.8532
0.0030
0.0042
0.0047
0.0040
20
10
0.8710
0.8601
0.8582
0.8611
0.0018
0.0024
0.0026
0.0023
15
0.8749
0.8678
0.8666
0.8684
0.0010
0.0013
0.0014
0.0012
P = 0.9
10
5
0.8593
0.8370
0.8315
0.8396
0.0052
0.0077
0.0092
0.0071
7
0.8660
0.8503
0.8470
0.8518
0.0032
0.0045
0.0051
0.0043
20
10
0.8725
0.8617
0.8598
0.8626
0.0016
0.0022
0.0024
0.0021
15
0.8766
0.8695
0.8684
0.8700
0.0009
0.0011
0.0012
0.0011


## 6. Conclusions

We develop some results on Rayleigh distribution when progressive type II censoring with binomial removals is performed. Based on the results shown in Table2, one can conclude that Bayesian estimators under precautionary loss function have the smallest biased and MSE as compared with the other estimators. The MSE associated with the estimators (Bayesian estimators under different loss functions and maximum likelihood estimators) of the parameter σ decrease with increasing the sample size n. According to Table 3 maximum likelihood estimators of reliability function of Rayleigh distribution have smallest biased and MSE as compared with the other estimators. It is immediate to note that MSEs of reliability function estimators decrease as n,m increases.



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*Source: 896807-2013-06-05.xml*