The comparison of empirical Bayes and generalized maximum likelihood estimates of reliability performances is made in terms of risk efficiencies when the data are progressively Type II censored from Rayleigh distribution. The empirical Bayes estimates are obtained using an asymmetric loss function. The risk functions of the estimates and risk efficiencies are obtained under this loss function. A real data set is presented to illustrate the proposed comparison method, and the performance of the estimates is examined and compared in terms of risk efficiencies by means of Monte Carlo simulations. The simulation results indicate that the proposed empirical Bayes estimates are more preferable than the generalized maximum likelihood estimates.
Censoring has become quite popular in life testing, reliability performances, and survival analysis. In life testing experiment, most of the time, an experimenter may be unable to observe the lifetimes of all units due to some constraints such as time, money, and resources, or it is not feasible to continue the experiment until the last observation. Censoring arises in such situations and different types of censoring can be used based on how the data are collected from the life testing experiments. The most popular censoring scheme among the various types of censoring scheme used in lifetime analysis is progressive Type II censoring scheme. Due to allowance of removal of experimental units at points other than the terminal point of an experiment, this censoring scheme is useful in many practical situations where budget constraints are in place or there is a demand for rapid testing. A significant work regarding the inference procedures based on progressive Type II censored samples has been found in the literature and journals, including Cohen [
In empirical Bayes (EB) approach, an experimenter usually does not specify the unknown hyperparameter (parameter of prior distribution) but is often tempted to use some estimate of the hyperparameter. This approach is commonly used when a datadriven choice of the hyperparameter is desirable and has been described extensively by many authors such as Robbins [
The main objective of the paper is to examine and compare the performance of the EB estimates of reliability performances relative to general entropy loss function (GELF) with that of the generalized maximum likelihood (GML) estimates in terms of risk efficiencies when the data are progressively Type II censored from the Rayleigh distribution.
The rest of the paper is organized as follows. Section
Rayleigh [
Let
It is clear that progressive Type II censoring scheme includes complete sampling scheme
It is straightforward to show that
The prior distribution is an integral part of Bayesian inference, which characterizes the beliefs of the researcher before observing the results of an experiment. Suppose that the unknown scale parameter
The prior distribution (
The GML estimate of the parameter
The invariance property of GML estimation enables one to obtain the GML estimates
Since the last couple of decades, a significant work has been developed in the field of reliability using EB approach. In this subsection, EB approach is used to derive the estimates of parameter
The sign and magnitude of the shape parameter
The Bayes estimates of reliability performances using GELF under progressive Type II censored data are, respectively,
Despite Bayes procedures that are increasingly popular, it is a common experience that expressing honest prior information about the parameter can be difficult and, in practice, one is often tempted to use some estimate of the hyperparameter. This mixed approach is usually referred to as empirical Bayes. Since the prior density (
Equation (
At this point, we obtain the risk functions of EB and GML estimates of reliability performances under GELF. The risk function of an estimate
The risk efficiency of an estimate
In this section, the performance of EB and GML estimates has been examined and compared on the basis of an extensive Monte Carlo simulation study and real life data application.
This subsection considers an extensive Monte Carlo simulation study to examine and compare the GML and EB estimates of reliability performances. The comparison is made on the basis of the risk efficiency criteria for different sample sizes (
Table
The risk efficiencies
As the effective sample proportion
The risk efficiencies
The risk efficiencies
The risk efficiencies under GELF are very sensitive to variation in “
Different values of the hyperparameter
Progressively Type II censoring schemes (C.S.) applied in the simulation study (

C.S. 


100  [ 
( 
[ 
( 



90  [ 
( 
[ 
( 



80  [ 
( 
[ 
( 



70  [ 
( 
[ 
( 



60  [ 
( 
[ 
( 
GML and EB estimates of parameter
C.S. 

 






[ 
0.15953  0.16132  0.16121  0.16083  0.16071 
[ 
0.14362  0.14511  0.14501  0.14470  0.14460 
[ 
0.16136  0.16319  0.16308  0.16269  0.16257 
[ 
0.14498  0.14649  0.14640  0.14608  0.14598 
[ 
0.16401  0.16590  0.16578  0.16538  0.16526 
[ 
0.14690  0.14845  0.14835  0.14802  0.14793 
[ 
0.16816  0.17012  0.17000  0.16959  0.16946 
[ 
0.14980  0.15141  0.15131  0.15097  0.15087 
[ 
0.18264  0.18488  0.18474  0.18426  0.18412 
[ 
0.15473  0.15643  0.15633  0.15597  0.15586 
Risk efficiency of
C.S. 
 






[ 
1.01499  1.01316  1.01300  1.01480 
[ 
1.01132  1.00885  1.00477  1.00474 
[ 
1.01542  1.01367  1.01400  1.01607 
[ 
1.01162  1.00920  1.00543  1.00552 
[ 
1.01604  1.01440  1.01548  1.01793 
[ 
1.01206  1.00972  1.00638  1.00668 
[ 
1.01699  1.01555  1.01783  1.02092 
[ 
1.01272  1.01049  1.00782  1.00843 
[ 
1.02027  1.01949  1.02642  1.03227 
[ 
1.01387  1.01184  1.01041  1.01160 
GML and EB estimates of reliability
C.S. 









[ 
0.96837  0.96866  0.96865  0.96865  0.96864 
[ 
0.96112  0.96145  0.96145  0.96143  0.96143 
[ 
0.96907  0.96936  0.96935  0.96934  0.96934 
[ 
0.96183  0.96216  0.96216  0.96214  0.96214 
[ 
0.97004  0.97032  0.97032  0.97031  0.97031 
[ 
0.96280  0.96312  0.96312  0.96311  0.96310 
[ 
0.97148  0.97175  0.97174  0.97174  0.97173 
[ 
0.96420  0.96452  0.96452  0.96450  0.96450 
[ 
0.97576  0.97600  0.97600  0.97600  0.97599 
[ 
0.96640  0.96670  0.96670  0.96669  0.96669 
Risk efficiency of
C.S. 
 






[ 
1.00505  1.00151  1.00017  1.00084 
[ 
1.00622  1.00213  1.00026  1.00096 
[ 
1.00498  1.00148  1.00016  1.00075 
[ 
1.00605  1.00202  1.00024  1.00086 
[ 
1.00488  1.00145  1.00014  1.00067 
[ 
1.00588  1.00192  1.00022  1.00077 
[ 
1.00473  1.00141  1.00011  1.00059 
[ 
1.00557  1.00172  1.00018  1.00068 
[ 
1.00430  1.00127  1.00009  1.00045 
[ 
1.00528  1.00159  1.00017  1.00051 
GML and EB estimates of failure rate
C.S. 









[ 
1.60797  1.59047  1.58611  1.57152  1.56714 
[ 
1.98391  1.96402  1.95906  1.94248  1.93749 
[ 
1.57169  1.55445  1.55014  1.53577  1.53145 
[ 
1.94688  1.92721  1.92230  1.90590  1.90097 
[ 
1.52142  1.50454  1.50033  1.48625  1.48202 
[ 
1.89643  1.87707  1.87223  1.85609  1.85124 
[ 
1.44761  1.01774  1.42718  1.41356  1.40946 
[ 
1.82369  1.80477  1.80005  1.78428  1.77954 
[ 
1.22733  1.21271  1.20906  1.19688  1.19322 
[ 
1.70965  1.69147  1.68693  1.67177  1.66722 
Risk efficiency of
C.S. 
 






[ 
1.02079  1.01620  1.01666  1.02375 
[ 
1.00559  1.00486  1.01328  1.02074 
[ 
1.02288  1.01765  1.01705  1.02409 
[ 
1.00670  1.00573  1.01357  1.02100 
[ 
1.02306  1.01980  1.01761  1.02457 
[ 
1.00836  1.00702  1.01397  1.02137 
[ 
1.02443  1.02328  1.01848  1.02532 
[ 
1.01092  1.00898  1.01458  1.02192 
[ 
1.02567  1.02690  1.02143  1.02780 
[ 
1.01571  1.01255  1.01563  1.02286 
We have considered the wind speed data set of Elanora Heights, located at northeastern suburb of Sydney, Australia [
We have checked the validity of the Rayleigh model based on the estimated value (moment estimate) of parameter
As a numerical illustration, we have generated artificial progressive and conventional Type II censored samples of size
The estimates and risk efficiencies for the progressive Type II censored data.










−0.8  1.15179  1.08914  0.99937  1.00004  0.03105  1.02717 
−0.5  1.14963  1.08625  0.99937  1.00003  0.03083  1.04094 
0.5  1.14256  1.06394  0.99937  1.00001  0.03007  1.09466 
0.8  1.14046  1.04728  0.99937  1.00005  0.02985  1.11571 
The estimates and risk efficiencies for the conventional Type II censored data.










−0.8  0.89914  1.12030  0.99898  1.00005  0.05093  1.04200 
−0.5  0.89748  1.11823  0.99898  1.00002  0.05057  1.06014 
0.5  0.89203  1.09030  0.99898  1.00003  0.04935  1.12661 
0.8  0.89041  1.06840  0.99898  1.00001  0.04898  1.15132 
From Tables
This paper is devoted to the comparative study of the performance of EB and GML estimates of reliability performances. We have obtained the GML estimates and EB estimates under GELF when the data are progressively Type II censored from the Rayleigh distribution. These estimates are then compared using the risk efficiency criteria. We have presented the real data example to illustrate the proposed estimation methods. A Monte Carlo simulation study is carried out to compare the performance of EB estimates with that of GML estimates. The study demonstrates that the EB estimates under GELF exhibit better performance than the GML estimates. Moreover, the risk efficiencies get smaller with the increasing ratio
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are thankful to the honorable editor and anonymous reviewers for valuable comments and constructive suggestions, which significantly improved the paper.