Exact Inference for an Exponential Parameter under Generalized Adaptive Progressive Hybrid Censored Competing Risks Data

: It is known that the lifetimes of items may not be recorded exactly. In addition, it is known that more than one risk factor (RisF) may be present at the same time. In this paper, we discuss exact likelihood inference for competing risk model (CoRiM) with generalized adaptive progressive hybrid censored exponential data. We derive the conditional moment generating function (ConMGF) of the maximum likelihood estimators of scale parameters of exponential distribution (ExpD) and the resulting lower conﬁdence bound under generalized adaptive progressive hybrid censoring scheme (GeAdPHCS). From the example data, it can be seen that the PDF of MLE is almost symmetrical.


Introduction
Let us consider a lifetime test where items are kept under observation until failure. These items could be patients put under certain medical research or they could be parts or some system in reliability test. However, it is known that the lifetimes of items may not be recorded exactly. There are also conditions wherein the elimination of items prior to failure is prearranged in order to lower the cost or time associated with test. In addition, it is known that more than one RisF may be present at the same time. Following Cox [1], we refer to this model as CoRiM. In CoRiM, it is supposed that the RisFs are statistically independent. Moreover, it is supposed that an indicator denoting the RisF of failure and competing risks (CoR) data consist of observed failure times. Lately, we are interested in one special cause in the presence of other RisFs.
Based on CoRiM, Author1 [2] developed CoRiM under generalized Type I hybrid censoring scheme and constructed exact confidence intervals (CI) and approximate CIs by exact distributions, asymptotic distributions, the parametric bootstrap method and the Bayesian posterior distribution, respectively.
Step stress partially accelerated life testing plan for competing risk using adaptive Type I progressive hybrid censoring (Ad1PHCS) was discussed by Author1 [3]. For Weibull distribution, Author1 [4] developed a CoRiM under progressive Type II censoring scheme (Pr2CS) with binomial removals. For Lindley distribution, Author1 [5] developed a CoRiM under Pr2CS with binomial removals. For Chen distribution, [6] developed a CoRiM under Pr2CS.
Conventional Type I and II censoring schemes cannot be used if the experimenter wants to remove the live experimental unit at a point other than the final end point of the experiment. Therefore, recently, Pr2CS and Ad1PHCS have become quite popular in a life-testing problem and reliability analysis [7][8][9][10][11]. Although Pr2CS and Ad1PHCS assure a pre-assigned number of failures, they have the drawback that it might take a long time to observe a pre-assigned number of failures and terminate In this paper, we consider independent identically distributed (iid) exponential CoRiM under GeAdPHCS. In Section 2, we derive the distributions of the MLEs of parameters as well as CIs for MLEs of parameters. In Section 3, we present the results of a numerical study to investigate the MSEs, biases, confidence lengths (CL) and coverage percentages (CP) of the MLEs under various GeAdPHCS.
From GeAdPHCS data, therefore, we have the following data: (x 1:m:n , z 1 ), (x 2:m:n , z 2 ), · · · , (x u:m:n , z u ), where u = m for Cases I and II and u = D 2 for Case III. Based on the three scenarios as discussed above, the likelihood function (L) is 1 + j x j:m:n + ∑ m j=d 1 +1 x j:m:n , Here, we denote the total failure number of units due to the RisF j by n j , j = 1, 2, and then it is easy to obtain n 1 = ∑ u i=1 z i and n 2 = ∑ u i=1 (1 − z i ) = u − n 1 . Note that, from (2), the MLEs do not exist when n j = 0, j = 1, 2. To estimate θ j , we have to observe at least one failure caused by each RisF. That is, Corollary 2. The first and second moments ofθ 2 are given by Theorem 3. Conditional on ξ (u) , the conditional PDF ofθ 1 is given by where γ(x − c; a, b) denote gamma distribution with shape parameter a, rate parameter b and shift parameter c.
Proof. The proof of Theorem 3 is given in Appendix B.

Corollary 3.
Conditional on ξ (u) , the tail probability ofθ 1 can be expressed as

Corollary 4.
Conditional on ξ (u) , the tail probability ofθ 2 can be expressed as Based on Corollaries 3 and 4, we construct 100(1 − α)% CI ofθ j under the assumption that P(θ j > k) is an increasing function of θ j , when the other parameter is fixed. Then, we can easily obtain the CI for θ j , denoted by (θ
Furthermore, if x m:m:n < T 1 , we have Case I and the corresponding GeAdPHCS data are {(x 1:m:n , z 1 ), (x 2:m:n , z 2 ), · · · , (x m:m:n , z m )}. If T 1 < x m:m:n < T 2 , we have Case II and the corresponding GeAdPHCS data are {(x 1:m:n , z 1 ), (x 2:m:n , z 2 ), · · · , (x m:m:n , z m )}, and d 1 = · · · = m−1 = 0. If T 2 < x m:m:n , we have Case III and we find D 2 such that x D 2 :m:n < T 2 < x D 2 +1:m:n . The corresponding GeAdPHCS data are (x 1:m:n , z 1 ), (x 2:m:n , z 2 ), · · · , (x D 2 :m:n , z D 2 ) . We reiterated the procedure 1000 times in each GeAdPHCS. We calculated the RMSEs of the estimator, and the corresponding average biases. The simulation results are presented in Table 1. In addition, we calculated the average CL and the corresponding CP. The results are presented in Table 2. Note that we used Python for the simulation study.   In Table 1, the following general observations can be made. The MSEs decrease as sample size n increases. For fixed sample size n, the MSEs decrease generally as the number of Pr2CS data size m increases. For Fixed sample size n and Pr2CS data size m, the RMSEs decreases generally as the time T 2 increases. In addition, we can observed that the estimator for Pr2CS I has smaller RMSE and bias than the corresponding estimator for the other two Pr2CS.

RMSE (Bias)
In Table 2, the CL decrease as sample size n increases. For fixed sample size n, the CL decrease generally as the number of Pr2CS data size m increases. For fixed sample size n and Pr2CS data size m, the CL decreases generally as the time T 2 increases. In addition, we can observed that the estimator for Pr2CS I has smaller CL than the corresponding estimator for the other two Pr2CS. It is observed that the CI works well for all GeAdPHCS. θ 1 has smaller RMSE, bias and CL than the correspondingθ 2 . This is because, when θ 1 is smaller than θ 2 , we may observe more failure number due to Factor 1 than those due to Factor 2, so that theθ 1 is more precise thanθ 2 .

Data Analysis
To analyze real data, we use the estimators in the above section. The real data were from some small electronic appliances exposed to the automatic test machine [14]. These data were analyzed by the authors of [2,15]. From these data, let us express the failure of appliance due to ninth failure RisF with z i = 1, and z i = 0 denotes failure caused by other failure RisFs. Here, we suppose that the underlying distribution of these data is the ExpD based on the Pr2CS (i.e., m = 28, 21 = · · · = 28 = 1 and i = 0 for i = 1, 2, · · · , 20). Then, Pr2CS data are presented in Table 3. Table 3. Progressive censored data of the example. In addition, we set Case I (T 1 = 7000 and T 2 = 8000), Case II (T 1 = 3000 and T 2 = 7000) and Case III (T 1 = 3000 and T 1 = 5000). Table 4 presents the 95% CIs forθ 1 andθ 2 , and we include the standard error (SE) and MSE calculated from Corollaries 1 and 2. In addition, the PDFs ofθ 1 andθ 2 based on the example data are shown in Figure 2.

Conclusions
It is known that the lifetimes of items may not be recorded exactly. Therefore, recently, Pr2CS and Ad1PHCS have become quite popular in a life-testing problem and reliability analysis. Although Pr2CS and Ad1PHCS assure a pre-assigned number of failures, it has the drawback that it might take a long time to observe a pre-assigned number of failures and terminate the test. For this reason, Lee [12] suggested a GeAdPHCS in which the test is assured to end at a pre-assigned time. In addition, it is known that more than one RisF may be present at the same time. That is, several RisFs compete for the immediate failure cause of items. Following Cox [1], we refer to this model as CoRiM. In this paper, we discuss exact likelihood inference for CoRiM with GeAdPHCS exponential data. We derive the ConMGF of the maximum likelihood estimators of scale parameters of ExpD and the resulting lower confidence bound under GeAdPHCS. Consequently, for fixed sample and Pr2CS sample size, the RMSEs decrease as the time T 2 increases. In addition, for fixed sample and Pr2CS sample size, the CLs decrease as the time T 2 increases. Although we focus on the inference for scale parameter of the ExpD, the suggested GeAdPHCS CoRiM can be extended to other distributions.

Conflicts of Interest:
The authors declare no conflict of interest.