Comparison of Some of Estimation methods of Stress-Strength Model: R = P(Y < X < Z)

In this study, the stress-strength model R = P(Y < X < Z) is discussed as an important parts of reliability system by assuming that the random variables follow Invers Rayleigh Distribution. Some traditional estimation methods are used   to estimate the parameters namely; Maximum Likelihood, Moment method, and Uniformly Minimum Variance Unbiased estimator and Shrinkage estimator using three types of shrinkage weight factors. As well as, Monte Carlo simulation are used to compare the estimation methods based on mean squared error criteria.


Introduction:
The many technological and mechanical malfunction problems that emerge due to the constant developments of life might not be the only reason behind the study of reliability and stress-strength models as a parts of reliability system since these models have several applications in different sciences areas, like reliability design of systems, quality control, physics, engineering and medicine. According to Rao 'the reliability can be defined as 'the probability of a device performing its function over a specified period of time and under specified operating conditions' (1). The probability equation: = ( < ) of "stress-strength" reliability shows that if the component which has a random strength represented by X is greater than the stress represented by the random variables Y where both of X and Y are independent then the system works statistically. Statistical studies of stress-strength reliability system were introduced for the first time in 1956 by Birnbaum (2). This paper is interested in the model R= ( < < ) that discusses the case where the strength should not only be greater than stress but also be smaller than stress , for example person's blood pressure has two limits systolic and diastolic and his/her blood pressure should lie within these limits (3). Singh in 1980, constructed maximum likelihood (ML), the minimum variance unbiased (MVU), and empirical estimators of stress-strength model = ( < < ) by assuming that strength of a component lies in an interval where , and are independent random variable based on normal distribution (4). After that in 2013 Wang et al. made statistical inference for R using nonparametric normal-approximations and the jackknife empirical likelihood under the assumption that the three samples were, independent, without ties among them (5). The Maximum likelihood estimator, moment estimator (ME) and mixture estimator (Mix) for the estimation of R=( < < ) and the stresses and and the strength X have Weibull distribution with common known shape and scale parameters and X, Y and Z are independent constructed in 2013 by Amal et al. (6). In 2016 Patowary et al. discussed the technique of Reliability estimation for P( < < ) of nstandby system (n=1, 2), through Monte-Carlo simulation (MCS) (7). In this work, the estimation of the system reliability P(Y < X < Z) based on the Inverse Rayleigh Distribution is considered. In addition, Monte Carlo simulation is performed for comparing several methods.

Model Description
Inverse Rayleigh Distribution (IRD) is first suggested by Trayer in 1964 see (8) and it has many applications in the area of reliability studies. The probability density function (pdf) and the corresponding cumulative distribution function (CDF) of one parameter Inverse Rayleigh Distribution are respectively defined as: where is the scale parameter for IRD see )9(. The Reliability and Hazard Rate functions of Inverse Rayleigh distribution are given, respectively as follows For more details about IRD see [10,11,12,13 and 14] This article discusses the estimation of reliability stress-strength model = ( < < ) in case of IRD. Let the random variables , and Z be independent and follow IRD with the scale parameters 1 , 2 , 3 respectively. the p.d.f of the strength X is ( 1 + 2 )( 1 + 2 + 3 )

Maximum Likelihood Estimation of
The Maximum likelihood method is an important and commonly, since it contained properties for good estimate and has invariant property . In this section, the MLE obtained for = ( < < ) throw the derivation of scale parameters 1 , 2 , 3 ,of the r.v s . , , as following: Let 1 , 2 , … . . be a random strength sample of size n with pdf. as in (eq.3) then let 1 , 2 , … . . 1 and 1 , 2 , … . . 2 be the random stresses with pdf. as in eq.4 and eq.5 respectively, the Maximum Likelihood function of the observed sample is : Taking the logarithm of likelihood function eq.7 then differentiating the result partially with respect to 1 , 2 , 3 and equalizing to zero respectively to get the estimated parameters ̂1,̂2 ,̂3 as follows: When substituting the equations 8, 9 and 10 in ̂ this leads to the estimation of the stress-strength model = ( < < ) using Maximum likelihood Estimator as bellow

Moments Method Estimator
This method is one of conventional methods because its ease. The basic idea of this method is to equate certain sample characteristics, such as the mean, to the corresponding population expected values. Then solving these equations for unknown parameter values yields the estimators. The sample mean of the r.v s . , , of IRD with unknown scale parameters 1 , 2 and 3 respectively are given by While ℎ moments will be: Equalize the sample mean with the first moment of X , Y and Z the estimates of 1 , 2 and 3 become: Substituted eq.12, 13and eq.14 in eq.6 to obtain the approximate moment estimator ̂ for stressstrength reliability ( ) as follows: ̂=̂1̂3 An unbiased estimator ̂of is called the uniformly minimum variance unbiased estimator (UMVUE) if and only if Var( ̂) ≤ Var( ̂) for any ∈ and any other unbiased estimator of α (15). The Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the scale parameters 1 , 2 and 3 of the random variables X ,Y and Z respectively of IRD will be discussed in this section. The IRD belongs to the exponential family with densities of the form then is a complete sufficient statistic for ( ) for i=1, 2,3..

Shrinkage Estimation Methods (Sh)
Thompson in 1968, suggested a new idea for estimate the unknown parameter by shrinking the classical estimator(̂) to a prior information(prior estimate) α 0 due the past studies or previous experiences using shrinkage weight factor Θ(̂) as below (16) Where ̂ℎ is the shrinkage estimator of and ̂ is unbiased estimator of as it is defined previously where ̂=̂( ) . The factor Θ(̂) ( 0 ≤ Θ(̂) ≤ 1 ) , represent a shrinkage weight factor which can be obtained by minimizing the mean squared error of ̂ℎ or it can be considered as a constant , a function of sample size or a function of ̂ ( ad hoc basis) . Noted Θ(̂) refers to the believe of unbiased estimator ̂, and (1 − Θ(̂)) represents to believe of 0 . Although the shrinkage estimator is biased, it is well known that it has minimum quadratic risk compared to classical estimators (mostly the maximum likelihood estimator or unbiased estimator); (17). In this section shrinkage estimator obtained according to Thompson's shrinkage technique also for more details see (18,19 and 20) . Where, i=1, 2 ,3 and refer to n , 1 2 or respectively depends on i. Thus, the shrinkage estimator of the scale parameters 1 , 2 , 3 of the random variables X,Y, Z that follows IRD will be as follows:

…(37) Shrinkage weight function (Sh wf )
In this subsection, the shrinkage weight factor has been suggested to be a function of n , m 1 and m 2 respectively in eq.33 as bellow Substitute equations 38,39and 40 in equation 6 to get the estimation of reliability models( ̂ℎ 2 ) using shrinkage function estimator as following :

Simulation
Monte Carlo simulation is used to compare the estimators obtained in this study. samples of different sizes, n = 10, 35, and 75 has been generated from one parameter inverse Rayleigh distribution based on MSE criteria, with 1000 replicates. The steps of Simulation for Mote Carlo as follows; Step1: Generate random samples as 1 , 2 , … , And, by the same method, the following equations have been gotten: Step3: Compute the R from equation (6).
Step4: Recall the R of the maximum likelihood estimator using equation (11).
Step4: Find Moment estimate for reliability by using equation (16).
Step4: Find Least Square Estimator estimate for reliability by using equation (32).
Step5: find the uniformly minimum variance unbiased method of R using equation (22).
Step7: Based on L=1000 replicate. Calculate the MSE as follows:

Results of Simulation
In this section, the simulation results used to determine the best outcome of the proposed estimation methods ( Based on the results, the shrinkage estimator ( ̂S h2 ) using Shrinkage weight function as shown in these tables was the best one and had less MSE after ̂ℎ in all cases for the = P( 1 < < 2 ). Therefore, Under the current results, it is observed that the results of ̂S h2 are better than the results of other approaches . .
-We hereby confirm that all the Figures and Tables  in the manuscript are mine ours. Besides, the  Figures and images, which are not mine ours, have  been given the permission for re-publication attached with the manuscript. -Ethical Clearance: The project was approved by the local ethical committee in University of Garmian.