Analysis of Generalized Exponential Distribution Under Adaptive Type-II Progressive Hybrid Censored Competing Risks Data

This paper presents estimates of the parameters based on adaptive type-II progressive hybrid censoring scheme (AT-II PHCS) in the presence of the competing risks model. We consider the competing risks have generalized exponential distributions (GED). The maximum likelihood method is used to derive point and asymptotic confidence intervals for the unknown parameters. The relative risks due to each cause of failure are investigated. A real data set is used to illustrate the theoretical results and to test the hypothesis that the causes of failure follow the generalized exponential distributions against the exponential distribution (ED).


Introduction
In the context of life testing experiments, hybrid censoring scheme was introduced at first by Epstein [3]. Since the introduction by Epstein [3], extensive works and different types of hybrid censoring scheme has been appeared. Recently, Kundu and Joarder [8] and Childs et al [2] both investigated the type-II progressive hybrid censoring scheme (T-II   . The drawback of the T-II PHCS is that the effective number of failures is random and it can be a very small number (even equal to zero), so that usual statistical inference procedures will not be applicable or they will have low efficiency. For this reason, Ng et al [14] suggested an adaptive type-II progressive hybrid censoring scheme in which the effective number of failures m is fixed in advance and the progressive censoring scheme 12 , ,..., m R R R is provided, but the values of some of the i R may be change accordingly during the experiment. Suppose the experimenter provides a time T, which is an ideal total test, but the experimental time is allowed to run over time T. If the m-th progressively censored observed failures occurs before time T (i.e. :: mmn XT  ), the experiment stops at this time :: mmn X , and we will have a usual type-II progressive censoring scheme with the prefixed progressive censoring scheme 12  . This formulate on leads to terminate the experiment as soon as possible if the (J+1)-th failure time is greater than T, and the total test time will not be too far away from the time T. If 0 T  , the scheme will lead us to the case of conventional type-II censoring scheme, and if T , we will have a usual progressive type-II censoring scheme. We should mention here that many authors studied the statistical properties of some life time models under AT-II PHCS in the presence of one and two causes of failures. Ng et al [14] developed inferential methods for the case when the lifetime distribution is exponential. They observed that the MLE always exists in this case. Lin et al [5], considered the adaptive progressive censoring scheme when the lifetime distribution is Weibull, and discussed the corresponding inferential issues. They have also discussed confidence intervals for the model parameters through the use of the asymptotic distribution of the MLEs as well as by the bootstrap method. Hemmati and Khorram [11], studied the competing risk model based on exponential distributions under the adaptive type-II progressively censoring scheme. They obtained the maximum likelihood and the Bayes estimators of the exponential distribution parameter, and two sides Bayesian probability intervals of the parameter are also obtained. Hemmati and Khorram [10], obtained the maximum likelihood estimators of the parameters from a two-parameter log-normal distribution based on the adaptive Type-II progressive hybrid censoring scheme, they compared the results with corresponding estimators of the type-II progressive hybrid censoring scheme. Mahmoud et al [15], obtained the maximum likelihood estimators for the unknown parameters of Pareto distribution based on the adaptive type-II progressive censoring scheme, point estimation and confidence intervals based on maximum likelihood also proposed. The main aim of this paper is analyzing the AT-II PHCS under the competing risk model when lifetimes have independent GED. We derive the maximum likelihood estimates (MLE) and we obtain the approximate two sided confidence intervals of these different parameters. We use the likelihood ratio test to test the ED against the GED. We consider a real data set and see how the different models work in the practical situation. The rest of this paper is organized as follows: In section (2), we introduce the model and the notation used throughout this paper. In section (3), we discuss the maximum likelihood estimation; confidence intervals are presented in section (4). In section (5), Goodness of fit test for testing a competing risks model where causes follow a GED against ED. In section (6), a real data set is used to illustrate the theoretical results.

Model description and notation
In reliability analysis, the failure of items may be attributable to more than one cause at the same time. These "causes" are competing for the failure of the experimental unit. Consider a life time experiment with nN  identical units, where its lifetimes are described by independent and identically distributed (i.i.d) random variables 12 [16] as generalization of the exponential distribution with the probability density function () where k  is the scale parameter and k  is the shape parameters. The cumulative distribution function () k Fx and failure hazard function () Under AT-II PHCS and in presence of competing risks data we have the following observation: where :: i i m n xx  for simplicity of notation, ( ) 1 and C is a constant doesn't depend on the parameters.

Maximum likelihood estimation
From (1), (2) and (4), the likelihood function ignoring the normalized constant can be written as follows The first order derivations of (6) with respect to k  and k  , 1,2 k  are given, respectively, by 3 where Equating the first derivations in (7) to zero, one can obtain the MLE of the unknown parameters 1 2 1 ,,    and 2  . As it seems, the system of non-linear equations (7) has no closed form solution in 1 Once 1  is computed, we determine 2  using the relation 21 1   . As the integral in the right side of (9) has no analytical solution, we have to use a numerical technique to solve the integral. According to the invariance property of the MLE, the MLE of the relative risk rates 1  , can be obtained by replacing the MLE of 1 2 1 ,,    and 2  in (9). Based on the above results, When 12 1   , the MLE's of 1  and 2  and the relative risk rates 1  and 2  , corresponds to the results of the exponential distribution obtained by Hemmati and Khorram [11], when the cause of failure is known.

Asymptotic confidence intervals
In this section we derive the confidence intervals of the vector of the unknown parameters  

Goodness of fit
We now discuss the problem of testing goodness of fit of a competing risks model when the causes of failures follow the GED against the ED to illustrate whether the GED can better fit a real data set rather than the ED studied by Hemmati and Khorram [11]. Because the ED can be derived as a special case of the GED, the likelihood ratio test will be used to test the adequacy of generalized exponential distributions competing risks. The null and alternative hypotheses are 0 where ED and GED are the log-likelihood functions under 0 H and 1 H , respectively, after replacing the unknown parameters with their MLE. For comparison purposes between the candidate models, we can use two model criterion selection, the Akaike information criterion (AIC) (Akaike [13]) and Bayes information criterion (BIC) (Schwarz [12]) defined as AIC 2 2 and BIC 2 .ln( ) where p is the number of parameters in the model, and is the maximized value of the likelihood function for the model. As a model selection criterion, the researcher should choose the model that minimizes AIC and BIC.

Numerical results
In this section, we analyze one data set which was originally analyzed by Hoel [6] and later by Kundu et al [7], Pareek et al [4], Cramer and Schmiedt [9], Hemmati and Khorram [11] and Ashour and Nassar [17]. The data was obtained from a laboratory experiment in which male mice received a radiation dose of 300 roentgens at 35 days to 42 days (5-6 weeks) of age. The cause of death for each mouse was determined by reticulum cell sarcoma as cause 1 and other causes of death as cause 2, there were m  and 2 18 m  . From the above data, the MLEs of the unknown parameters , the corresponding approximate 95% two sided confidence intervals distributions, the log-likelihood values ( ), AIC and BIC shown given in table (1).  , and the p-value is 0.00003. The analysis of the previous real data set demonstrates the importance and usefulness of adaptive type-II progressive hybrid censoring scheme and inferential procedures based on them. From example 1 and 2, it is observed that T plays a major role in the estimation and for the construction of the corresponding confidence intervals, because when T increases some additional information is gathered. We also conclude that based on the values of L  , the p-value, AIC and BIC, the GED fits the data better than ED. We have observed that the assumptions that the generalized exponential distributions may be used to analyze this set of real data better that the exponential distribution.