Estimation of the Stress-Strength Reliability for Exponentiated Pareto Distribution Using Median and Ranked Set Sampling Methods

In reliability analysis, the stress-strength model is often used to describe the life of a component which has a random strength (X) and is subjected to a random stress (Y). In this paper, we considered the problem of estimating the reliability R = P [Y < X] when the distributions of both stress and strength are independent and follow exponentiated Pareto distribution. The maximum likelihood estimator of the stress strength reliability is calculated under simple random sample, ranked set sampling and median ranked set sampling methods. Four different reliability estimators under median ranked set sampling are derived. Two estimators are obtained when both strength and stress have an odd or an even set size. The two other estimators are obtained when the strength has an odd size and the stress has an even set size and vice versa. The performances of the suggested estimators are compared with their competitors under simple random sample via a simulation study. The simulation study revealed that the stress strength reliability estimates based on ranked set sampling and median ranked set sampling are more efficient than their competitors via simple random sample. In general, the stress strength reliability estimates based on median ranked set sampling are smaller than the corresponding estimates under ranked set sampling and simple random sample methods.

used simple random sample (SRS) method based on the same number of measurement units. The RSS can be described as follows: Step 1: Randomly select 2 m units from the population of interest and randomly allocated them into m sets, each of size m.
Step 2: The m units within each set are ranked by visual inspection or by any cheap method concerning the variable of interest.
Step 3: The smallest ranked unit is measured from the first set of m units; the second smallest ranked unit is measured from the second set of m units. The process is continued until the largest ranked unit is measured from the last set.
Step 4: The entire process can be repeated several times; say r times, to produce an RSS sample of size n=rm. Muttlak [Muttlak (1997)] proposed median ranked set sampling (MRSS) as a modification of the RSS to estimate the population means. The MRSS technique can be summarized as follows: Select m random samples each of size m units from the study population. Rank the units within each set concerning a study variable. If the sample size m is odd, from each sample select for measurement the   ( 1) / 2 m th  smallest ranked unit, i.e., the median of the sample. If the sample size m is even, select for the measurement from the first ( 2) m samples the   2 m th smallest ranked unit and from the second ( 2) m samples the   ( 2) 1 m th  smallest ranked unit. The cycle can be repeated r times if needed to get a sample of size n=rm units from the MRSS data. For more details about RSS, Hassan [Hassan (2012)] considered the goodness of fit tests for the exponentiated Pareto distribution using an extreme RSS method. Hassan [Hassan (2013)] obtained the maximum likelihood and Bayesian estimators based on RSS. Haq et al. [Haq, Brown, Moltchanova et al. (2014)] introduced ordered double RSS. Haq, et al. [Haq, Brown, Moltchanova et al. (2015)] introduced the varied L RSS scheme as a modification of the RSS. Al-Omari [Al-Omari (2015)] considered the estimation of the distribution function based on L ranked set sampling. Al-Omari [Al-Omari (2016)] proposed a new measure of sample entropy of continuous random variables in RSS. Zamanzade et al. [Zamanzade and Al-Omari (2016)] proposed a new generalization of the RSS for estimating the population means and variance. Al-Omari et al. [Al-Omari and Zamanzade (2017)] proposed the goodness of-fit-tests for the Laplace distribution based on RSS. Syam et al. [Syam, Al-Omari and Ibrahim (2017)] investigated the mean estimation using stratified double MRSS. Bouza et al. [Bouza, Al-Omari, Santiago et al. (2017)] considered the ratio type estimation using the knowledge of the auxiliary variable for ranking and estimating. Al-Omari et al. ] introduced new estimators of the population ratio using multistage MRSS. Al-Nasser et al. ] suggested a new sampling method called MiniMax RSS for estimating the population parameters. Al-Omari et al.  proposed some entropy estimators of a continuous random variable using RSS. Zamanzade [Zamanzade (2019)] developed a non-parametric cumulative distribution function estimator based on pair RSS. Al-Omari et al. [Al-Omari and Haq (2019 b)] suggested a new sampling method for estimating the population mean. Zamanzade et al. [Zamanzade and Wang (2020)] investigated nonparametric estimation using partially ordered sets. Also, see ; Al-Nasser and Al-Omari (2015); Santiago, Bouza, Sautto et al. (2016)]. Making statistical inferences concerning the reliability =P [Y < X] have received considerable attention in the context of the reliability. In this sense, X represents the strength of a system that is subjected to stress Y. It is clear that the system fails when the stress exceeds strength. Therefore, the stress-strength (SS) model is a measure of system reliability. In general, statistical inference about the SS model is considered based on SRS data. However, in recent years, statistical inferences about the SS model based on the RSS method have been considered by several researches. Sengupta et al. [Sengupta and Mukhati (2008)] discussed estimation of the SS reliability for exponential populations. Muttlak et al. [Muttlak, Abu-Dayyeh, Saleh et al. (2010)] proposed three estimators of the SS reliability when X and Y are independent one-parameter exponential populations. Hassan et al. [Hassan, Assar and Yahya (2014); Hassan, Assar and Yahya (2015)] discussed the estimation of the SS reliability when Y and X are two independent Burr type XII distribution under several modifications of RSS. Estimation of the SS model for Weibull distribution has been discussed by Akgül et al. [Akgül and Şenoğlu (2017)]. Akgül et al. [Akgül, Acıtaş and Şenoğluc (2018)] discussed the estimation of the SS reliability when X and Y independent Lindley populations. The Pareto distribution is well-known in the literature due to its capability in modeling the heavy-tailed distributions. Applications of the Pareto distribution appear in several areas including economics, finance, stock price fluctuations, environmental studies, insurance risk, etc. Recently, it has also been recognized as a useful model for the analysis of lifetime data. Gupta et al. [Gupta, Gupta and Gupta (1998)] proposed a simple generalization of the well-known standard Pareto distribution of the second kind, named as the exponentiated Pareto (EP) distribution. A two-parameter EP distribution can have decreasing and upside-down bathtub shaped failure rates. The probability density function (pdf) of the EP distribution is given by where θ and  are two shape parameters. The cumulative distribution function of the EP distribution is defined as The EP distribution reduces to the standard Pareto distribution of the second kind when θ=1. The studies about the EP distribution have been provided by some authors. Parameter estimation of the EP is considered by Shawky et al. [Shawky and Abu-Zinadah (2009)]. Estimation of the reliability in a multi-component SS model following EP distribution, for identical and non-identical components, were discussed by Hassan et al. ; Hassan and Basheikh (2012 b)]. Chena et al. [Chena and Cheng (2017)] considered the estimation of the SS model for EP populations. Estimation in a partially accelerated life test for the EP distribution based on progressive censoring has been provided by Hassan et al. [Hassan, Abd-Alla and El-Elaa (2017)]. Parameter estimators of the EP distribution in presence of outliers have been considered by Nooghabi [Nooghabi (2017)]. The ML, least squares, weighted least squares and Bayesian estimators of an extended EP distribution were discussed by Hassan et al. [Hassan, Nassr and Hemeda (2020)].
In this study, we obtained the estimators of the SS reliability based on SRS, RSS and MRSS schemes. It is assumed that the strength ~EP ( , )  and the stress ~EP ( , )  are both independent. The main reason for using the EP distribution is that its flexibility for modeling several lifetime data and its extensive consideration in engineering, life testing and reliability studies. A simulation study is conducted to compare the behavior of the suggested different estimates. The rest of this article is organized as follows. Section 2 provides the explicit expression of the SS reliability as well as the ML estimator of the reliability R which is obtained under SRS. Section 3 gives the ML estimator of the SS reliability based on RSS. Section 4 comes up with four different reliability estimators based on the MRSS method. The numerical study is provided in Section 5. Eventually, the paper is concluded in Section 6.

Reliability estimator based on SRS
This section provides an explicit expression of the SS reliability. The ML estimator of the reliability is derived via SRS.

Stress strength reliability
Let X is the strength of a system and Y is the stress acting on it. Let X and Y are the two independent strength-stress random variables observed from EP ( , )  and EP ( , )  respectively. The SS reliability parameter is evaluated under the assumption that the models have the same shape parameter  but with different shape parameters  and  that is; EP ( , )  and EP ( , ).  Therefore, the SS reliability parameter R is given by If  and  are known, then the reliability parameter is simply calculated using Eq. (3). It can be observed that the SS reliability does not depend on .  However, if  and  are unknown and  is known, the estimators of  and  depend on ,  and hence so does the estimator of the SS reliability.

Reliability estimator based on SRS
Suppose that X 1 , X 2 , . . . , X n is a random sample from EP ( , )  and Y 1 , Y 2 , . . . , Y m is a sample from EP ( , ).
 The ML estimator of the reliability given that the sample is obtained. To compute the ML estimator of the reliability, it is required to obtain the ML estimators of  and .
 The joint log-likelihood function for the observed samples is Setting Eqs. (5), (6) and (7) with zeros we obtain the ML estimators of ,  and .  From Eqs. (5) and (6), we obtain the ML estimator of  and  as a function of  as follows:

If
 is known, the ML estimators of  and  can be obtained from Eqs. (8) and (9). In case of all the parameters are unknown, then  can first be estimated by solving the following nonlinear equation: Since  is a fixed point solution of the nonlinear Eq. (10), hence, it can be obtained by using a simple iterative procedure. Therefore, the ML estimator of the SS reliability, denoted by ˆ, R can be obtained as follows

Reliability estimator based on RSS
To obtain the ML estimator of SS reliability; suppose {X i(i)c , i =1, 2…, n; c=1, 2, …, r x ) be an RSS observed from X~EP ( , )  , with sample size nr x , where n is the set size and r x is  with sample size mr y , where m is the set size and r y is the number of cycles.
Therefore, the likelihood functions L 1 of the observed data will be as follows: are the pdfs of random variables X i(i)c and Y j(j)d , respectively, where () 0 The log-likelihood function of L 1 will be as follows: The ML estimators of ,  and  can be obtained by maximizing ln L 1 directly with respect to ,  and  .
The first partial derivatives of the log-likelihood function with respect to ,  and  are given by: Clearly, it is not easy to obtain a closed-form solution to non-linear Eqs. (16), (Error! Reference source not found.) and (18), so we apply an iterative technique to solve these equations numerically. Hence, we obtain the ML estimators of ,  and ,  then by using invariance property the ML estimator of the reliability is obtained by using Eq. (3).

Reliability estimator based on MRSS
The MRSS has been proposed by Muttlak [Muttlak (1997)] as a sampling technique to estimate the population mean. The MRSS procedure depends on two cases, the first one for odd set size while the second case for even set size. In this section, an estimator of the SS reliability is obtained when both stress and strength have odd or even set sizes. Additionally, the ML estimator of SS reliability is obtained when strength X has an odd set size and stress Y has an even set size and vice versa.

Reliability estimator with odd set size
In this sub-section, the ML estimator of the SS reliability is provided when both X and Y are drawn using MRSS with an odd set size. Let X i(g)c ; i=1,…,n, c=1,…,r x ; g=  ( 1) 2 n  be an MRSS observed from X~ EP ( , )  , with sample size nr x , where n is the set size and r x is  with sample size mr y , where m is the set size and r y is the number of cycles. Therefore, the likelihood function L 2 of the observed data will be as follows: The loglikelihood function of L 2 will be as follows: The ML estimators of ,  and  can be obtained by maximizing ln L 2 directly with respect to ,  and .  The first partial derivatives of ln L 2 with respect to ,  and  are given by: Since Eqs. (23), (24) and (25) have no closed-form solution, so an iterative technique is employed to solve these equations numerically. Hence, the ML estimators of the unknown parameters based on the MRSS in case of an odd set size are obtained. Therefore, the ML estimator of the SS reliability is obtained by substituting the ML estimators of population parameters in Eq. (3).

Reliability estimator with even set size
Here, the reliability estimator is investigated when both X and Y are drawn based on MRSS with even set size. In case of the even set sizes, the qth smallest unit is chosen from the first q ordered sets, while the (q+1)th smallest unit is chosen from each of the Therefore, the likelihood functions L 3 of the observed data will be as follows: where H 3 is a constant, ( 1)ln 1 1 , (  (37) and (38) are solved numerically via iterative technique to obtain the ML estimators of population parameters. Further, the ML estimators of population parameters will be employed in Eq. (3) to get the reliability estimator.

Reliability estimator with odd strength and even stress set sizes
Here, the reliability estimator is obtained when the strength of X has MRSS with an odd set size, while the stress of Y has MRSS with an even set size. Let X i(g) where H 4 is a constant, ( ) As it seems the likelihood Eqs. (45), (46) and (47) have no closed-form solutions. Therefore, the numerical technique will be utilized to get the solution. The ML estimators of population parameters will be employed in Eq.
(3) to get the reliability estimator.

Reliability estimator with even strength and odd stress set sizes
Here, the reliability estimator is obtained when the strength of X is selected using MRSS with even set size, while the stress of Y is selected based on MRSS with an odd set size.  with sample size mr y , where m is the set size and r y is the number of cycles. Therefore, the likelihood functions L 5 of observed data, where X and Y independent, is given by:

Let the set {X
Therefore, the likelihood functions L 5 of the observed data will be as follows:   and  can be obtained by maximizing ln L 5 directly with respect to ,  and .  The first partial derivatives of log-likelihood function with respect to ,  and  are given by: ( ) ( 1) 1 1 1 ln 1 ( 1)ln 1 ln and It is clear that the Eqs. (48), (49) and (50) do not have an analytical solution, so the ML estimators of ,  and  can also be obtained by numerical methods. Hence, the ML estimators of  and  are employed in Eq.

Simulation study
In this section, an extensive simulation study is provided to compare the performance of the reliability estimates based on RSS and MRSS with their SRS counterparts. In this comparison, the absolute bias, the mean square error and the relative efficiency criteria are utilized. In the simulation setup, the set sizes and the number of cycles are selected as             Fig. 7 shows that the efficiency of the reliability estimate under MRSS is greater than the efficiency of the reliability estimate under RSS expect at (m x , m y )=(3,3),(5,5),(7,7), where the true value of R =0.6.  The reliability estimate under MRSS has greater efficiency than the reliability estimate based on RSS in case of the true value of R= 0.714 (see Fig. 8).  For the true value R = 0.833, the reliability estimate under MRSS has greater efficiency than the reliability estimate under RSS for all values of (m x , m y ) expect at (2,3).  As the set size increases, the efficiency of all the reliability estimates increases in almost all cases.

Conclusions
In this article, the estimation of the reliability R =P[Y<X] is studied when the strength X and stress Y are independent variables follow the exponentiated Pareto distribution. The maximum likelihood estimators of R are computed using SRS, RSS and MRSS. Based on MRSS, the reliability estimate is considered in four different cases. The simulation study is performed to evaluate the performance of the different proposed estimates. From the simulation study, it is found that, the MSEs of the reliability estimates based on SRS data are greater than their competitors based on RSS and MRSS data, respectively. The MSEs of the reliability estimates under MRSS are the smallest in most of the cases as compared with the corresponding estimates using RSS and SRS data. The efficiency of all reliability estimates increases as the set size increases in almost cases. Also, it is found that the reliability estimates using MRSS is more efficient than the reliability estimates based on RSS. Also, the reliability estimates based on RSS and MRSS are more efficient than the reliability estimates under SRS. In general, the reliability estimates under MRSS are more efficient as compared with the other reliability estimates based on RSS and SRS methods