Parameters Estimation for the Exponentiated Weibull Distribution Based on Generalized Progressive Hybrid Censoring Schemes

Abstract Based on Type-I and Type-II generalized progressive hybrid censoring schemes, the maximum likelihood estimators and Bayes estimators for the unknown parameters of exponentiated Weibull lifetime model are derived. The approximate asymptotic variance-covariance matrix and approximate confidence intervals based on the asymptotic normality of the classical estimators are obtained. Independent non-informative types of priors are considered for the unknown parameters to develop the Bayes estimators and corresponding Bayes risks under a squared error loss function. Proposed estimators cannot be expressed in closed forms and can be evaluated numerically by some suitable iterative procedure. Finally, one real data set is analyzed for illustrative purposes.


Introduction
Many studies have considered censored samples which are Type-I, Type-II, hybrid and progressive hybrid censoring scheme. The exponentiated-Weibull distribution has been utilized for the analysis of life testing and reliability data. Recently, progressive hybrid censoring schemes (PHCSs) have become quite popular in a life testing problems and reliability analysis. Kundu and Joarder [6] proposed a PHCS, which is a mixture of Type-II progressive and hybrid censoring schemes, Childs et al. [2] refer to this censoring scheme as a Type-I PHCS and proposed Type-II PHCS to overcome the obvious drawback of the Type-I PHCS, that is maximum likelihood estimator (MLE) may not always exist. Type-I and Type-II generalized progressive hybrid censoring schemes (GPHCSs) proposed to overcome the drawbacks of the Type-I PHCS and Type-II PHCS, respectively. Cho et al. [3] proposed a new censoring scheme called GPHCS to overcome the drawback of the Type-I PHCS. One limitation of the Type-I PHCS, that is cannot be applied it when very few failures may occur before time T accordingly the MLE for a parameter of underling lifetime model of observations may not be computed or its accuracy will be extremely low. Therefore, Cho et al. [3] suggested this Type of censoring to allow the experiment to continue beyond time T and observed a pre-specified number of failures if very few failures had been observed up to time .
T Under GPHCS the experimenter would ideally like to observe -th m failures, but is willing to accept a bare minimum of -th k failures. Lee et al. [8] refer to the GPHCS as a Type-I GPHCS and proposed the Type-II GPHCS to overcome the drawbacks in Type-II PHCS is that it might take a very long time to observe -th m failures and complete the life test. Type-II GPHCS is a modified for Type-II PHCS by guaranteeing that the test will be completed at time 2 T , therefore, 2 T represents the absolute longest time that the researcher is willing to allow the experiment to continue. They suggested this type of censoring scheme to a guarantee the experiment terminated at a pre-fixed time.
Some recent studies on Type-I and Type-II GPHCSs have been carried out by many authors, including, Cho et al. [3] considered the Bayesian and maximum likelihood estimations for the entropy of Weibull distribution based on Type-I GPHCS. Cho et al. [4] obtained the exact distribution of the MLE as well as exact lower confidence bound for the exponential parameter under Type-I GPHCS.
Lee et al. [7] obtained the MLE for the unknown parameter of exponential distribution under Type-II GPHCS. They derived to the exact and approximate conditional inference for the proposed estimator under Type-II GPHCS. Lee et al. [8] derived to exact inference of the unknown parameters under the assumptions that the lifetime distributions of different causes are independent identically distributed (IID) exponential competing risks model under Type-I GPHCS. Ashour and Elshahhat [1] obtained the MLEs and Bayes estimators for the unknown parameters of Weibull distribution based on Type-II GPHCS as well as they developed Bayes estimates and Bayes risks under a square error loss (SEL) function.
The rest of the paper is organized as follows: In Section 2, Type-I and Type-II GPHCSs are described. In Section 3, the MLEs are introduced for the unknown parameters of exponentiated Weibull (EW) distribution under Type-I and Type-II GPHCSs as well as some special cases are given, approximate asymptotic variance-covariance (V-Cov) matrix and approximate confidence interval (CI) for the MLEs using asymptotic distribution are obtained. In Section 4, the Bayes estimators and Bayes risks are developed under a SEL function and using independent non-informative priors for the unknown parameters of EW distribution based on Type-I and Type-II GPHCSs. It is clear that, the Bayesian and non-Bayesian estimators for EW parameters are not in closed forms, therefore, in Section 5, one numerical example is considered to illustrate the proposed estimators by using MathCad package version 14. Finally, we conclude the paper in Section 6.

Model Description and Notation
Type-I and Type-II GPHCSs are proposed to overcome the drawbacks of the Type-I and Type-II PHCSs, respectively, and they can be described as follows  Figure 1.

Type-I GPHCS
Based on the Type-I GPHCS, the observed data will be one of the following three forms: The likelihood function of the Type-I GPHCS can be written in the following form where, 1, 2, 3, i = for Case-I,II and III, respectively,  T , as a schematic illustration is depicted in Figure 2.

Type-II GPHCS
Based on the Type-II GPHCS, the observed data will be one of the following three forms: The likelihood function of the Type-II GPHCS can be written in the following form where, 1, 2, 3, i = for Case-I, II and III, respectively, 1 1 ,

Maximum Likelihood Estimators
EW family introduced by Mudholkar and Srivastava [9], which is a simple generalization of well-known twoparameter Weibull distribution and is obtained by introducing one additional shape parameter. The EW distribution has been applied in areas of reliability analysis, quality control, duration and failure time modeling. Mudholkar et al. [10] presented and illustrated some applications of the EW distribution in reliability and survival studies. Mudholkar and Hutson [11] illustrate usefulness of the EW distribution in modeling extreme value data using the floods and Nassar and Eissa [12] derived to expressions for the mode of the EW distribution.
Suppose that the observed failures are IID from threeparameter

American Journal of Applied Mathematics and Statistics
and CDF is ( ) where, α and θ both are the shape parameters and β is the scale parameter.
The EW distribution has a two very well-known lifetime distributions as a special cases, if 1 α = and 1 θ = , the EW distribution (3) reduced to exponential distribution with β is scale parameter and if 1 θ = , the EW distribution reduced to Weibull distribution with twoparameter α is shape parameter and β is scale parameter.
Assuming that the failure times of the experimental units are follows the three-parameter EW distribution with PDF (3) and CDF (4), then the MLEs of the EW parameters can be obtained under Type-I and Type-II GPHCSs, respectively, as follows:

MLEs of EW Parameters Based on Type-I GPHCS
Based on the PDF and the CDF of EW distribution (3) and (4), respectively, then the likelihood function of the Type-I GPHCS (1) will be ( ) Additionally, the corresponding log-likelihood function of (5) can be written as follows American Journal of Applied Mathematics and Statistics Equating the first derivations (7) to zero and solving for , α β and θ , we get the MLEs , α β and θ of , α β and θ based on Type-I GPHCS, respectively, in the following forms Clearly, the MLEs , α β and θ of EW parameters , α β and θ based on Type-I GPHCS, respectively, can be obtained by solving set of nonlinear equations, this needs computer facilities and numerical techniques. Also, Cho et al. [4] results can be obtained as a special case from (6), if putting 1 α θ = = , we get the MLE β in the case of exponential parameter β , if putting 1 θ = , we get the MLEs α and β in the case of Weibull parameters α and β , respectively.

MLEs of EW Parameters Based on Type-II GPHCS
Based on the PDF and the CDF of EW distribution (3) and (4), respectively, then the likelihood function of the Type-II GPHCS (2) can be rewritten as follows: for Case-II, Additionally, the corresponding log-likelihood function of (8) will be ( ) where, 1, 2, 3, log , , Differentiating (9) with respect to , α β and θ , respectively, we get Equating the first derivations (10) to zero and solving for , α β and θ , we get the MLEs , α β and θ of , α β and θ based on Type-II GPHCS, respectively, as in the following forms Clearly, the MLEs , α β and θ of EW parameters , α β and θ based on Type-II GPHCS, respectively, do not result in closed forms, this needs computer facilities and numerical techniques to evaluated numerically. Some special cases can be obtained from (9), Lee et al. [7] results in the case of exponential distribution by putting 1 α θ = = . Also, Ashour and Elshahhat [1]) results in the case of Weibull distribution by putting 1 θ = .
Based on the log-likelihood functions (6) and (9), the approximate asymptotic V-Cov matrix for the MLEs of the three-parameter EW distribution can be obtained based on Type-I and Type-II GPHCSs, respectively. The elements of the observed information matrix (11) are obtained based on the log-likelihood functions of Type-I GPHCS (6) and Type-II GPHCS (9) and reported in Appendix A and B, respectively.
Under the regularity conditions for the asymptotic properties of MLEs of EW parameters α , β and θ , the asymptotic normality of the MLEs ˆi α , ˆi β and ˆi θ is approximately multivariately normal, i.e.,

Bayes Estimators
Following Singh et al. (2005), the Bayes estimators can be develop for the EW parameters α , β and θ based on Type-I and Type-II GPHCSs, we consider independent non-informative priors 1 ( ) g α , 2 ( ) g β and 3 ( ) g θ , given as

Bayes Estimators of EW Parameters Based on Type-I GPHCS
Based on the likelihood function (5), the noninformative priors (12) and using the Bayes theorem, the joint posterior distribution of ω given data X can be written with proportional as follows the normalizing constant 1 ψ of (13) is given by

A T ω =
for Case-III.
Marginal posterior densities of the EW parameters , α β and θ can be obtained, respectively, by integrating (13) with respect to the other two parameters as follows ( ) ( ) and ( ) .
Based on the SEL function and the marginal PDF of , α β and θ as in (14), (15) and (16) respectively, the Bayes estimators , α β   and θ  of the EW parameters , α β and θ , respectively, becomes ( ) ( ) Clearly, the Bayes estimators and the corresponding Bayes risks of the EW parameters , α β and θ based on Type-I GPHCS do not result in closed forms due to involvement of multidimensional integrals which are not solvable analytically. Therefore, the Bayes estimates and the Bayes risks of , α β   and θ  can be evaluated by using computer facilities and numerical techniques.

Bayes Estimators of EW Parameters Based on Type-II GPHCS
Based on the likelihood function (8), non-informative priors (12) and using the Bayes theorem, the joint posterior distribution of ω given data X can be written with proportional as follows the normalizing constant 2 ψ of (17) is given by for Case-III.
Marginal posterior densities of , α β and θ obtained by integrating (17) with respect to the other two parameters as follows and ( ) .
Based on the SEL function and the marginal PDF of , α β and θ as in (18)

Real Data Analysis
Previous sections dealt with the analytical technique and this section focuses on the numerical one through practical data set, which was originally presented by Nichols and Padgett [13]. This data set was obtained from a process producing carbon fibers to be used in constructing fibrous composite materials. The ordered data with 100 n = observations on breaking stress of carbon fibers (in Gba) are in Table 1. One question arises about whether the data fit the EW distribution or not. To check for the goodness of fit, we compute the chi-square test. First, the EW distribution will be fitting using the MLEs and then carrying out chi-square goodness of fit test. The MLEs Using the MathCad package and the real data set as in Table 1 where, F is the CDF of the EW distribution, U Y and L Y are the lower and upper limits for class i and n is the sample size. The observed and the expected frequencies of the ordered data set can be calculated and reported in Table 2: To obtain the maximum likelihood estimates and Bayes estimates for the EW parameters under Type-I and Type-II GPHCSs, the progressively Type-II censored sample will be proposed in a design under Type-I and Type-II GPHCSs as in Table 3.
Notice that, in Table 3, (-) represents to a number of observed failures at time which is less than the termination point of the life test.
All computations were performed using MathCad package version 14. The maximum likelihood estimates and approximate CIs for the unknown parameters of EW distribution α , β and θ based on Type-I and Type-II GPHCSs are calculated and reported in Table 4 and Table  5, respectively. To evaluate the Bayes estimators, some various values of hyper-parameter 1, 2 c = and 4 are considered. The Bayes estimates and the corresponding Bayes risks of the unknown EW parameters based on Type-I and Type-II GPHCSs are reported in Table 6 and  Table 7, respectively.

Conclusions
In this paper, we have considered the Bayes and non-Bayes estimations for the unknown parameters of the EW distribution based on Type-I and Type-II GPHCSs. Some special cases using exponential and Weibull distributions are obtained, i.e., based on Type-I GPHCS, Cho et al. [4] results were generalized in the case of exponential and Weibull distributions at 1 α θ = = and 1 θ = , respectively. Also, based on Type-II GPHCS, Lee et al. [7] results in the case of exponential parameter can be obtained as a special case at 1 α θ = = as well as if putting 1 θ = , Ashour and Elshahhat [1] results can be obtained as a special case in the case of Weibull distribution. The MLEs, approximate V-Cov matrix and the approximate CIs based on the observed Fisher information matrix have been discussed. Independent non-informative priors are considered to provide the Bayes estimators and the corresponding Bayes risks under the SEL function. Based on Type-I and Type-II GPHCSs, the MLEs and the Bayes estimators for EW parameters do not result in explicit forms, therefore, a numerical example has been presented to illustrate all the inferential results established here. As expected, Table 4 and Table 5 showed that the maximum likelihood estimates for the unknown parameters of EW distribution based on both Types of GPHCSs are more precise than the Bayes estimates as in Table 6 and Table 7. Therefore, if prior information of the EW parameters is not available, then it is always better to use the MLEs rather than the Bayes estimators, because the Bayes estimators are computationally more expensive.