Constant Stress Partially Accelerated Life Tests for Extended Generalized log Logistic Distribution Based on Type I Censored Competing Risks Data

: As a result of technology improvement getting information about products and materials lifetimes under usual conditions. Therefore accelerated life testing or partially accelerated life testing usually are used to truncate the tests survives. The test items under accelerated life testing run under accelerated conditions and partially life tests run under both accelerated and use conditions. The main idea of accelerated life testing that the acceleration element is not unknown or the mathematical model relating the lifetime of the unit and the stress is known or can be assumed. In some cases, neither acceleration factor nor life-stress relations are not unknown. This paper concerned with studying and discussed the constant–stress partially accelerated life test (CPALT) under type I censored (T.I.C) competing risks data. Failure times resulting from T.I.C competing risks data are assumed to follow the Extended generalized log logistic (EGLL) distribution because this model is completely flexible to study positive data. This distribution is applied in various fields, for example lifetime studies, economics, finance and insurance. The maximum likelihood (ML) method is used to estimate the parameters under TIC competing risks data. The simulation algorithm is performed to assess the theoretical results of the maximum likelihood estimates based on TIC competing risks


Introduction
In experiments ALT is applied to reduce time and cast. There are different methods of acceleration. Among these methods, the constant stress in which stress on the units remains constant, the progressive stress in which the stress applied on the test units increases with time and the step stress in which the test conditions change for a given time or for a given number of failures, see DS. Bai, SW. [1] In PALT the test items are run at both accelerated and normal conditions. PALT is suitable when the acceleration failure are the mathematical model is unknown, see Abdel-Hamid and Al-Hussaini [2], Hassan et.al. [3], Hassan et.al. [4], Abu-Zinadah and Ahmed [5], Ismail [6], A.A. Ismail, A.A. Albabtain [7], Ismail and Al Tamim [8], Ismail and Al Harbi [9], Li and Zheng [10] , Zarrin, el al. [11], Fawzy [12], The EGLL was first introduced by Lima and Corderio [13].
This new model is very flexible for modeling various types of data. In this paper, the MLES of the EGLL parameters are obtained under constant stress partially accelerated life test. The rest of this paper is organized as follows: Competing risks schemes and model description are presented in Section 2. The ML estimators under T.I.C competing risks data is illustrated in Section 3. The simulation study is performed to assess the theoretical results in Section 4

Competing Risks Plans and Model Description
In Survival study, the items failure might be credit ready to more reason simultaneously. Theses "causes" would contending for the test unit failure. In the statistical literature, this issue is common as those contenting risks pattern. The information of competing risks study, the data comprises of a time failure and the related cause of its failure. It is assumed that the failure reasons are independent or reliant. In this study, we expect the suppressed failure time experiment, as proposed by Cox [14], that the times failure are individually circulated. where the failure is caused because of many reasons of failure, see Crowder [15].

Model Description and Its Assumptions
In this section, we show the fundamental assumptions for the life test of the product in CPALT competing failure of the model. Additionally, the procedures of test in CS_PALT dependent on T.I.C plans that the competing failures lifetimes are expected to be EGLL distribution are clarified.
The procedure of test in CPALT is recognized as: 1) Whole n units are split into two sets: 2) Group1(G.1) consists of = (1 − ), (1 − ) is sample part units shared to typical circumstances. 3) Group 2 (G.2) consists of = residual units are exposed to accelerated conditions. 4) All items in G.1 and G.2 are run at steady stress level till the test ends after the time of censoring τ in the case of T.I.C is reached. 5) The lifetimes , = 1,2, … , (1 − ), of units designated in ordinary states are EGLL distribution with shape parameter !, " , scale parameter # and its probability density function (pdf) and cumulative distribution function (cdf) are given as follows: and where the observed ordered failure times are And  The probability work for T.I.C contending risks of data at when time the reason of failure is known at ordinary circumstances as follows:

Simulation Study
A simulation study is used to assess the estimates performance. The assessments of the acceleration factor (q , q ) and population parameters ! , " , # , ! , " , # are evaluated regarding the mean squared errors (MSEs) and biases. The numerical technique is approached as follows: 1   Table 1. The outcomes can be found as follows: 1) The Abias and MSEs decrease as n increase under T.I.C data.
2) It is noticed that the Abias and MSEs decrease, censoring time ; decrease. 3) Clearly, the acceleration of the experiment is useful to get outcomes and data quickly, yet the most consequences of normal condition are more exact that speeding up condition.

Conclusion
In this study, the CPALT under T.I.C data is discussed assuming that failure times are EGLL distribution. The ML method is applied to estimate the unknown parameters under TIC competing risks. To assess the theoretical results of the ML method for CPALT based on T.I.C data, the simulation algorithm is performed. It is noted that the Abias and MSEs decrease as n increase under T.I.C data and censoring time ; decrease. The acceleration of the experiment is valuable helpful to get outcomes and data quickly, yet the most consequences of normal condition are more exact that speeding up condition.