1. Introduction
In life-testing and reliability studies, both type I and type II censoring schemes are widely used. These two types of censoring schemes are described as follows: Consider
n identical components are placed in the test, in type I censoring, the experiment continues up to a predetermined time
. However, in the type II censoring scheme, the experiment is terminated when a predetermined number of failures
occurs. For a mixture of type I and type II censoring schemes, which is called the type I hybrid censoring scheme, is introduced by Epstein [
1], and the life test experiment is terminated at a random time
. Childs et al. [
2] proposed a new hybrid censoring scheme called a type-II hybrid censoring scheme in which the experiment would terminate at the random time
. These schemes do not allow for removing the components from the experiment at any time other than the terminal point. A more general censoring scheme called progressive type II censoring is used to deal with this problem.
In the progressive type II censoring, n components are placed in a lifetime testing experiment, and m is a fixed number of components to be failed. At the time of the first failure , components are randomly removed from the remaining components. Similarly, at the time of the second failure , components of the remaining components are randomly removed, and so on. At the time of the m-th failure , all the remaining components are removed. The are fixed and predetermined prior to the study.
The type I progressive hybrid censoring scheme is considered by Kundu and Joarder [
3], which is a mixture of type II progressive and hybrid type I censoring schemes—in which
n identical components are put under testing with
, predetermined progressive censoring scheme and the experiment is terminated at random time
. The experiment stops at the time
, if the
m-th failure occurs before the time
. However, if the
m-th failure does not occur before time
and only
J failures occur before the time
; then, at the time
, the experiment terminates and all the remaining components are removed. In addition, a type II progressive hybrid censoring scheme is discussed by Childs et al. [
4], where the experiment is terminated at time
.
A new censoring scheme, called an adaptive type II progressive hybrid censoring, is introduced by Ng et al. [
5], where the number of failures
m and the corresponding progressively scheme is given, but no components will be removed when the experimental time passes time
; see, for example, Balakrishnan and Kundu [
6]. Another adaptive progressive hybrid censoring scheme, called the adaptive type I progressive hybrid censoring scheme (AT-I PHCS), which assures the termination of the lifetime testing experiment at a fixed time
, and results in a higher efficiency in estimations, is proposed by Lin and Huang [
7]. The AT-I PHCS can be described as: suppose
n identical components are placed under testing with prefixed
,
; the experiment is terminated at a prefixed time
, where
. At the time
,
of the remaining components are randomly removed, at the time
,
of the remaining components are randomly removed, and so on. Let the number of failures that occur before time
be
J. If the
m-th failure
occurs before time
, the process will not stop, but continue to observe failures without any further withdrawals until reach time
. Then, all remaining components
are removed at time
, and the experiment is terminated. The progressive censoring scheme in this case will become
, where
. Otherwise, when
, the process will have a progressive censoring scheme such as
. AT-I PHCS is important when the time is the main goal in the experiment, and it is requisite to terminate the experiment at a predetermined time in any case of number of failures.
The point and interval estimation for the exponential distribution are studied by Lin and Huang [
7] and investigated Bayesian sampling plans under different progressive censoring schemes. The maximum likelihood and Bayesian estimation for a two-parameter Weibull distribution based on AT-I PHCS are discussed by Lin et al. [
8]. They derived the Bayes estimates of the unknown parameters by using the approximated form of Lindley [
9] and Tierney and Kadane [
10].
On the other hand, the loss function is important in Bayesian methods. In the Bayesian inference, the most commonly used loss function is the squared error loss. It is well known that the use of symmetric loss functions may be inappropriate in many circumstances, particularly when positive and negative errors have different consequences. One of the most commonly used asymmetric loss functions is the LINEX (linear exponential) loss function. It was introduced by Varian [
11] became popular due to Zellner [
12].
The failure of components in reliability analysis at the same time may be attributable to more than one reason. These reasons are competing for the experimental component for the failure. This is known as the competing risks model. The data in the competing risks analysis are comprised of a failure time and the associated reason for failure. The reasons for failure can be assumed to be independent or dependent.
The latent failure times model in this paper is assumed as suggested by Cox [
13]. In addition, the failure times are independently distributed. The failure is due to more than one reason, see Crowder [
14]. The competing risks data discussed here under AT-I PHCS and Weibull distributions with common shape parameters are assumed to the failure times. The competing risks data under AT-I PHCS with the assumption of exponential distribution are analyzed by Ashour and Nassar [
15]. The exponential distribution has a constant failure rate, so it has serious limitations in modeling lifetime data. The inference for Weibull distribution under adaptive type-I progressive hybrid censored competing risks data are investigated by Ashour and Nassar [
16].
The main aim of this paper is studying the competing risk model under AT-I PHCS. The lifetimes under the competing risks have independent Weibull distributions with common shape parameters. This paper can be organized as follows:
The model description and the notation are introduced in
Section 1. The maximum likelihood estimation of the unknown parameters is established in
Section 3. Bayesian estimation of the parameters under squared error (SE) and LINEX loss functions are discussed in
Section 4. The expected Bayesian estimation under squared error and LINEX loss functions are derived in
Section 5. Some properties of the E-Bayesian estimators are also derived in
Section 6. Finally, two examples of the real data set and numerical simulation results are presented in
Section 7.
2. Model Description
Suppose
n identical components are put into a lifetime test with prefixed progressive censoring scheme
and the experiment is terminated at time
where
. The lifetime for the components are assumed to be the Weibull distribution. Under the adaptive type-I progressive censoring scheme and in the presence of competing risks data, we have the following observation:
where the indicator
is denoting the reason of failure, and
J is the number of failures before time
.
Consider that
is the number of remaining components left at the time point
with
. Let
, here,
means that the component
i has failed due to reason
k. Furthermore, we define
Thus, the random variables describe the number of failures due to the reason for failure.
The latent failure times
and
are assumed to be independent.
, have Weibull distributions with parameters
and
(same shape and different scale parameter). The corresponding survival function
and the hazard rate function
are given, respectively, by
and
4. Bayesian Estimation
In this section, we obtain the Bayes estimators of the parameters
and
based on SE and LINEX loss functions. For developing the Bayesian estimation, we assume that the parameters
and
are independently distributed according to gamma distribution. Let
have a gamma prior with scale parameters
and shape parameters
. The joint prior density of
and
can be written as follows:
The joint prior (
9), can be derived as a special case from the dependent prior proposed by Kundu and Pradhan [
17]. Based on the likelihood function (
6) and the joint prior density (
9), the joint posterior density of
and
given
, and the data can be written as
From (
10), we observe that the posterior density functions of
and
are
and
, respectively. Based on the SE loss function, the Bayes estimators of
and
can be obtained as the posterior means with the following forms:
Considering
, the Bayes estimators in (
11) coincide with the MLEs in (
7).
Now, we obtain the Bayes estimators based on LINEX loss function proposed by Varian [
11]. The LINEX loss function with parameter
is given by
where
is a constant,
is the parameter to be estimated, and
is the Bayes estimator under the LINEX loss function. From (
12), the Bayes estimator
is
From (
10) and (
13), the Bayes estimators of the parameters
and
can be obtained as
Under the SE loss function, the MSE of the parameters,
,
, can be obtained as
In addition, when
, the Bayesian estimator,
,
, has the following properties:
5. E-Bayesian Estimation
E-Bayesian (Expected Bayesian) estimation was first introduced in literature by Han [
18]. He obtained the estimate of the scale parameter of the Weibull distribution based on SE loss function and also derived the properties of the E-Bayesian estimation. E-Bayesian based on three different prior distributions of hyper parameter are used in this section to investigate the influence of different prior distributions on the E-Bayesian of
. For more relevant research about the E-Bayesian estimation, see Han [
19], Jaheen and Okasha [
20], Azimi et al. [
21], Okasha [
22], Okasha and Wang [
23], and Abdallah and Jumping [
24].
Han [
18] stated that the prior distribution of
and
should be determined to ensure that the prior distribution
is a decreasing function in
. To be sure from this condition, we find the first derivative of
with respect to
as
Thus, for
and
, the function
and therefore
is a decreasing function of
. Suppose that
and
are independent with bivariate PDF given by
Then, the E-Bayesian estimate of the parameter
, (expectation of the Bayesian estimate of
), according to Han [
18] can be obtained as follows:
where
is the Bayesian estimator of
given by Equations (
11) and (
14),
is the expected Bayes estimate of
,
under any loss function. The value range of hyper parameter
and
satisfy
. Suppose the prior distribution of
and
are beta distribution and uniform distribution in
, respectively. For more details about E-Bayesian estimation, see Han [
18], Okasha [
22] and Okasha, and Wang [
23].
The expected mean square error (E-MSE) of the parameter
according to Han [
19] can be obtained as follows:
where
is the MSE of
,
under any loss function.
5.1. E-Bayesian Estimation under SE Loss Function
Based on three different prior distributions of the hyper-parameters
and
, the E-Bayesian estimates of the parameter
can be obtained. Accordingly, these prior distributions are selected to show the effect of the different prior distributions on the E-Bayesian estimation of the parameter
. The selected priors distributions are given by
These prior distributions are used to guarantee that
is a decreasing function in
. Now, under SE loss function, the E-Bayesian estimates of the parameter
can be obtained from (
11), (
17), and (
19). Using the prior distribution,
is given by
Similarly, under SE loss function, the E-Bayesian estimates of
based on
and
are given, respectively, by
and
The E-MSE of the parameter
can be obtained from (
15), (
18), and (
19). Using the prior distribution,
is given by
Similarly, under SE loss function, the E-MSE of
based on
and
are given, respectively, by
and
5.2. E-Bayesian Estimation under LINEX Loss Function
Under LINEX loss function, the E-Bayesian estimation of
can be obtained by using the different prior distributions of the hyper-parameters given by (
19). For the prior distribution,
and, based on (
14), (
17), and (
19), the E-Bayesian estimate of
is obtained as
Similarly, under LINEX loss function, the E-Bayesian estimates of
using
and
are given, respectively, by
and
The E-MSE of the parameter
can be obtained from (
16), (
18), and (
19). Using the prior distribution,
is given by
where
,
j = 1,2, is the Bayesian estimator of
given by Equation (
14),
is the mean square error of Bayesian estimator of
given by Equation (
16) and
D is the domain of
and
for which the prior density is decreasing in
.
The E-MSE of the parameter
can be obtained from (
14), (
18), and (
19). Using the prior distribution,
is given by
where
,
is the polylogarithm function,
and
Similarly, under LINEX loss function, the E-MSE of
based on
and
are given, respectively, by
6. Properties of E-Bayesian Estimation Based on SE Loss Function
Now, the relations among and E-MSE, (, ) estimations will be discussed.
I. Relations among, (, )
Proposition 1. Let , , and , (, ) be given by (20)–(22). Then, the following inequalities are - (i)
.
- (ii)
.
II. Relations among, (, )
Proposition 2. Let , , and , (, ) be given by (26)–(28). Then, the following inequalities are - (i)
.
- (ii)
.
III. Relations among E-MSE, (, )
Proposition 3. Let , , , and E-MSE () be given by (23)–(25). Then, the following inequalities are - (i)
.
- (ii)