Multicomponent Inverse Lomax Stress-Strength Reliability

In this article we derive two reliability mathematical expressions of two kinds of s-out of -k stress-strength model systems; and . Both stress and strength are assumed to have an Inverse Lomax distribution with unknown shape parameters and a common known scale parameter. The increase and decrease in the real values of the two reliabilities are studied according to the increase and decrease in the distribution parameters. Two estimation methods are used to estimate the distribution parameters and the reliabilities, which are Maximum Likelihood and Regression. A comparison is made between the estimators based on a simulation study by the mean squared error criteria, which revealed that the maximum likelihood estimator works the best.


1-Introduction
The Inverse Lomax distribution (ILD) is used in various fields such as stochastic modeling, economics, actuarial sciences and life testing; it is one of the notable lifetime models in statistical applications. The distribution belongs to an inverted family of distributions and found to be very flexible to analyze the situation where the non-monotonicity of the failure rate has been realized. If a random variable Y has Lomax distribution, then( X =1/Y) has an Inverse Lomax distribution [1][2][3][4][5].
The probability density function (pdf) and cumulative distribution function (cdf) of the random
, …(1) ) = Here and are the shape and scale parameters, respectively. Now onwards, Inverse Lomax distribution (ILD) with its two parameters will be denoted by ILD ( ). The purpose of this paper is to study two kinds of the reliability form of the multicomponent stress-strength models, based on X, Y being two independent random variables, where X~ ILD( ) and Y~ ILD( ). Stress-strength model is a system to analyze the strength of materials on which stress is, whatever the material type is; in either part or all of the system, the system collapses if the stress applied to it exceeds the strength, which has many applications in physics and engineering topics. This model is of a special importance in reliability literature [7]. For one component, let Y be a strength random variable subjected to a stress random variable X, where X and Y are independent, then the reliability of this system is ∫ Here we consider the problem of reliability estimation in a multicomponent stress-strength system (s-out of-k), which was studied by Bhattacharyya and Johnson [8]. Let the random variables X, Y 1 ,…,Y k be independent, F(x) be the continuous distribution function of X, and F(y) be the common continuous distribution function of Y 1 ,…,Y k, , then the reliability of the multicomponent stressstrength system is given by : In 2012, Hassan and Basheikh [7] expanded the system reliability of multicomponent to more than two groups of components. Consider a system made up of k non-identical components. Out of these components, are of one category and their strengths are independent and identical distribution random variables. The remaining components are of a different category and their strengths are independent identically distributed (iid) random variable subjected to a common stress X which is an independent random variable.
The reliability estimation for the two kinds of ILD multicomponent stress-strength systems has not received much attention in the literature. Therefore, an attempt is made here to study the reliability estimations for the two models. In section 2, we derive the mathematical expression for and . The maximum likelihood and the regression estimators for the ILD parameters which are used to give the estimators of and are obtained in section 3. The comparison between the two estimators of reliabilities for different experiments and sample sizes is made through a Monte Carlo simulation study in section 4. Finally, some conclusions and comments are provided in section 5.

and for Inverse Lomax distribution (ILD)
The reliability functions are obtained by using the probability in equation (2) for and of two multicomponent stress-strength models. Considering the multicomponent stress-strength system given by k components, where the strength with the distribution function (cdf) is given as: Under common stress with density function (pdf) given in eq. (1), then the reliability can be obtained by (1) and (3) in (2), where: The derivative of is: By transformation and substituting u and in (4), we get: Since and through the rearrangement of the boundaries, then the is given by Where are integers. Now to obtain the reliability , consider a system made up of k non-identical components. Let be an independent stress random variable, are independent identically distributed (iid) strength random variables, and the remaining are iid strength random variables with (cdf), given as respectively, then by (1), (6) and (7), the reliability , can be given as: By the same transformation technique, we get: ∑ …(10) Then by substitution (9) and (10) in (8), we get: Karam et al.

Estimation of the Inverse Lomax Distribution (ILD)
Estimation the reliability functions and is performed by using the maximum likelihood and the regression methods.

Maximum likelihood Estimation (MLE)
Let the y 1 , y 2 ,…, n strength sample has distribution with a sample size n, where is an unknown parameter, then the likelihood function L can be written as follows [9]: ∏ Then the likelihood function using the equation (1) And the estimated formula for the unknown stress parameter can be formulated as: By substituting the equation (24) and (25) in the equation (5), the approximate RM estimator for , , can be obtained as: In the same manner, we obtain the estimate for the unknown shape parameters for , , then the result will be : By Substituting the equations (27) and (28) in the equation (11) of the RM estimator for , , ̂ , we obtain:

Simulation study
In this section, we found the real values of the reliability in the two models discussed previously in section 3, at different values for the distribution parameters of the variables and for the case of two different experiments for s and k. The results are recorded in Tables-(1 and 2). It is observed from those two tables that for fixed strength parameters, the values of the reliabilities are increased by increasing the values of the stress parameter, and at the same value of stress parameter, the values of the two reliabilities are decreased by increasing the strength parameter. .7, as well as for different parameter values and for two different S out of K systems. As in the following steps: Step 1-Using MATLAB 2017 by generating random values of the random variables and by applying the inverse function as follows: Step 2-By computing the mean by the equation: Step 3-The comparison between the estimation methods by using MSE criteria: MSE= ∑ ̂ Where N is The number of replications for each experiment, which is 500. The results of comparison are shown as in Tables-3, 4, 5, and 6. From those tables, we found that for all experiments and for each of the small (15), moderate (30) and large (90) sample sizes, the preference is for the maximum likelihood estimators

Conclusions
In this paper, we study the multicomponent stress-strength reliability for the ILD with different shape parameters. After the implementation of simulation experiments, we found that the best estimation MSE for the maximum likelihood estimator works quite well for all experiments and for all sizes of samples, while the regression estimation was not performing well in the ILD of stress and strength.