Statistical Inference For A Simple Step–Stress Model With Type–II Hybrid Censored Data From The Kumaraswamy Weibull Distribution

In reliability and life–testing experiments, the researcher is often interested in the effects of extreme or varying stress factors on the lifetimes of experimental units. In this paper, a step–stress model is considered in which the life–testing experiment gets terminated either at a pre–fixed time (say, 1 m T + ) or at a random time ensuring at least a specified number of failures (Say, y out of n ). Under this model in which the data obtained are Type–II hybrid censored, the Kumaraswamy Weibull distribution is used for the underlying lifetimes. The maximum Likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model are derived. The confidence intervals of the parameters are also obtained. The hazard rate and reliability functions are estimated at usual conditions of stress. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.


Introduction
In order to obtain highly reliable products long life spans, consuming and expensive tests are often required to collect enough failure data. The standard life testing methods are not appropriate in such situations, and to overcome this difficulty accelerated life tests are applied; where in the test units are run at higher stress levels (which includes temperature, voltage, pressure, vibration, cycling rate and etc.) to cause rapid failures. Accelerated life tests (ALT) allow the experimenter to apply sever stresses to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Such tests can reduce the testing time and save a lot of man power, material sources and money. The stress can be applied in different ways: commonly used methods are constant stress, progressive stress and step-stress [Nelson (1990), Bagdonavicius and Nikulin (2002)].
In step-stress ALT, the stress for survival units is generally changed to a higher stress level at a predetermined time. This model assumes that the remaining life of a unit depends only on the current cumulative fraction failed and current stress [Lydersen and Rausand (1987)]. Moreover, if it is held at the current stress, survivors will continue failing according to the cumulative distribution function (CDF) of that stress but starting at the age corresponding to previous fraction failed. This model is called the Cumulative Exposure (CE) model. Some references in the failed of the accelerated life testing include [Jones (2009)]. See also for more details Hassan et al. (2016) and Chunfang and Yimin (2017).
Constructed a distribution with two shape parameters on (0,1). Kumaraswamy (KUM) distribution is applicable to many natural phenomena whose outcomes have lower and upper bounds, such as heights of individuals; scores obtain in a test, atmospheric temperatures and hydro logical data. Also, Kum distribution could be appropriate in situations where scientists use probability distribution which has infinite lower and or upper bounds to fit data, when in reality the bounds are finite. A compound between Kum distribution and any distribution was constructed by [Cordeiro, et. al. (2010)].
Weibull distribution is one of the most popular models; it has been extensively used for modeling data in reliability, engineering and biological studies. The need for forms of Weibull distribution arises in many applied areas. In this paper, simple step-stress is applied to Kumaraswamy Weibull (KUMW) distribution. The cumulative distribution function (CDF) and probability density function (pdf) of KUMW distribution are obtained as follows;  and  are the shape parameters,  is the scale parameter. It has three shape parameters, these parameters allows for highly degree of flexibility. Some special cases can be obtained from KUMW distribution such as KUM exponential, KUM Rayleigh, Weibull and exponential. It is wide applicable in reliability, engineering and in other areas of research. For more details about the KUMW distribution see

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The reliability function (Rf) and the hazard rate function (Hrf) corresponding to (1) can be written, respectively as follows; In this article, a step-stress model is considered in which the life testing experiment gets terminated either at a prefixed time (say, 1 m T + ) or at random time ensuring at least a specified number of failures (say, r out of n ). Under this model in which the data obtained are Type-II hybrid censored, the case of two stress levels is proposed with underlying lifetimes being KUMW distributed. The model considered here is discussed in detail in section (2). The statistical inference for simple step-stress life testing based on type-II hybrid censored is obtained in section (3).

Model Description
Assuming K step-stress ALT, the model of constant stress is considered in the first step. In this model, the lifetime of the unit is affected by a certain level of stress 1 S , where 1 S is larger than the usual stress 0 S . In the consecutive steps, other stresses are considered as 23  , the pdf of the KUMW ( , , , )     distribution can be written as follows; 2. ,,    are constants with respect to the stress S and the shape  parameter is affected by the stress , 1, 2,..., Gx is the population cumulative distribution on fraction of units failing under stepstress, then in the first step; By taking the logarithm for two sides, it follows that; The cumulative exposure model for j steps can be written as follows; (11), it is seen that () Gx , for a step-stress pattern which consists of segments of the CDF, 12 , ,..., k F F F , can be written in the form; and the associated pdf,has the following form;, () gx

Cumulative Exposure Model:
The life time distribution at 1 S and 2 S from (12) and (13); () Gx and () gx , can be written in the form; ( and the associated density functiontten in the form;is wri () gx .

Inference Simple Step-Stress Accelerated Life Tests Based on
,, With the notation, we will observe one of the following three cases for the observations: we will observe that; ... ...
The failure time distribution is assumed to be KUMW distribution and the shape parameter  is shown as a function of the stress through the inverse power law model.

Maximum Likelihood Estimation:
From the CED in (14) and the corresponding pdf in (15), the likelihood function of , , , cb and  based on the type-II HCS is obtained as follows; In this situation the log likelihood function of , , , cb and  is given by;  The first derivatives of the logarithm of likelihood function (18) with respect to, , , , cb and  are obtained. Therefore, the MLE can be obtained by equating the first derivatives of () nL to zero. As shown they are nonlinear equations, the estimates ˆ垐 , , , cb and  are numerically using Newton Raphson method [Chenhua (2009)  is a mission time. The asymptotic Fisher-Information Matrix can be written as follows;

The Confidence Interval (CIs) Based on Type-II HCS:
Based on large sample size, the MLEs under appropriate regularity conditions are consistent and asymptotically unbiased as well as asymptotically normally distributed. Therefore, the two side approximate 100 (1 ) where ˆw  is the standard deviation and this study ŵ is ĉ or b or  or  and or  , respectively [Nelson (1982)].

Numerical Results
This section aims to investigate the precision of the theoretical results of both estimation and optimal design plans on basis of simulated and real data.

Simulation Study:
Several data sets are generated from KUMW distribution for a combination of the initial parameters values of , , , cb and  for sample sizes 25, 50, 75, 100 and 125 using 1000 replications for each sample size. The transformation between uniform distribution and KUMW distribution in step Computer program is used depending on "Mathcad 14" using Newton Raphson method to solve the derived nonlinear logarithmic likelihood equations simultaneously.  (20) and (21).
The performance of ˆ垐 , , , cb and  has been evaluated through some measurements of accuracy. In order to study the precision and variation of MLEs, it is convenient to use the root mean squares error. The coverage probabilities and length for , , , cb and  , Root Mean Squares Error (RMSE), Absolute Value of Reliability Error (ARE) and Relative Bias (RB) will be obtained. The different sample size of 25,50, 75,100 n = and 125 are considered. These sample sizes are chosen to present the small, moderate and large sizes. The results are displayed in tables (1) to (3).
It is clear from table (1) that, the MLEs are much closed to the initial values of the parameters as sample size increases. Also, as shown in the numerical results RMSE, ARE and RB are decreasing when the sample size is increasing. For all sample sizes the following are observed; − , cband  performs better than  and  .
-141-− b and  performs better than ĉ . − b performs better than ĉ .  Table (2) indicates that; the reliability decreases when the mission time to increases. The results get better in the sense that the aim of an accelerated life testing experiments is to get large number of failure (reduce the reliability) of the device with high reliability. As to and when the sample size increase, the rf increase. The hrf increases when the mission time to increase. The two sides 95% central asymptotic CIs for the parameters of KUMW are displayed in table (3). This table contains the lower bound (L), upper bound (U), length of the intervals and the coverage probabilities. The interval estimate of the parameters becomes narrower as the sample size increases. For all sample sizes, it is clear that; -143-− The length of the interval for  is shorter than the other lengths. − The length of the interval for c is shorter than the length of the interval b . − Among CIs seems to have considerably low coverage (especially  ) compared to the nominal level. The coverage probabilities better and closer to the nominal level in general.

Illustrative Example:
In this subsection, the main aim is to demonstrate how the proposed method can be used in practice. Using Kolomogrov-Smirnov goodness of fit test where the data points representing failure time. The data were taken from [Murthy, et. al. (2004).] and analyzed by [Rezk1et al. (2014)], the data were 20 items (n=20) tested with test stopped after 20 th failure ( 20) The Kolomogrov-Smirnov test shows that the KUMW distribution provides a good fit to the data. It is assumed that 2 k = , i.e. there are only two different levels of stresses 1 (20). In this case, we had a fixed time 2

Summary and Conclusions
In this paper, a simple step-stress model with two stress levels from KUMW distribution is considered where the data are type-II HCS. The MLEs of the unknown parameters ˆ垐 , , , cb and  are derived. A simulation study is conducted to compare the performance of all these procedures. It is observed that the approximation method of constructing CIs (based on the asymptotic normality of MLEs, , , , cb and  ) is satisfactory in terms of coverage probabilities (quite close to the nominal level) for the parameters , , , cb and  . An example is presented to illustrate all methods of inference discussed here as well as to support the conclusions drawn. As a future work, one can estimate the KUMW distribution parameters under simple step-stress model based on some recently introduced schemes like joint progressive Type-I censored scheme, . Abo-Kasem and Nassar (2019).

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The results obtained in this paper can be modified to obtain results for sub-models of