Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

Inference of stress-strength for the Type-II generalized logistic distribution under progressively Type-II censored samples

This article studies the estimation of the reliability R = P[Y < X] when X and Y come from two independent generalized logistic distributions of Type-II with different parameters, based on progressively Type-II censored samples. When the common scale parameter is unknown, the maximum likelihood estimator and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Bayes estimator of R and the corresponding credible interval using the Gibbs sampling technique have been proposed too. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are extracted. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real dataset is given for illustrative purposes. Finally, methods are extended for proportional hazard rate models.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Inferences on stress–strength parameter based on GLD5 distribution

Let X and Y be independent random variables distributed as generalized Lindley distribution type 5 (GLD5). This article deals with the estimation of the stress–strength parameter R = P(Y < X), which plays an important role in reliability analysis. For this purpose, the maximum likelihood and the uniformly minimum variance unbiased estimators are presented in the explicit form. Moreover, considering Arnold and Strauss’ bivariate Gamma distribution as an informative prior and Jeffreys’ as noninformative prior, the Bayes estimators are derived. Various bootstrap confidence intervals are also proposed and, finally, the presented methods are compared using a simulation study.     

Estimation of δ=P(X<Y) for Burr XII distribution based on the progressively first failure-censored samples

Let X and Y have two-parameter Burr XII distributions. The maximum-likelihood estimator of δ=P(X<Y) is studied under the progressively first failure-censored samples. Three confidence intervals of δ are constructed by using an asymptotic distribution of the maximum-likelihood estimator of δ and two bootstrapping procedures, respectively. Some computational results from intensive simulations are presented. An illustrative example is provided to demonstrate the application of the proposed method.     

Stress-Strength Reliability of a Two-Parameter Bathtub-shaped Lifetime Distribution Based on Progressively Censored Samples

Based on progressively Type-II censored samples, this article deals with inference for the stress-strength reliability R = P(Y < X) when X and Y are two independent two-parameter bathtub-shape lifetime distributions with different scale parameters, but having the same shape parameter. Different methods for estimating the reliability are applied. The maximum likelihood estimate of R is derived. Also, its asymptotic distribution is used to construct an asymptotic confidence interval for R. Assuming that the shape parameter is known, the maximum likelihood estimator of R is obtained. Based on the exact distribution of the maximum likelihood estimator of R an exact confidence interval of that has been obtained. The uniformly minimum variance unbiased estimator are calculated for R. Bayes estimate of R and the associated credible interval are also got under the assumption of independent gamma priors. Monte Carlo simulations are performed to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose. Finally, we will generalize this distribution to the proportional hazard family with two parameters and derive various estimators in this family.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Inference of R=P[X<Y] for generalized logistic distribution

This paper deals with the estimation of R=P[X<Y] when X and Y come from two independent generalized logistic distributions with different parameters. The maximum-likelihood estimator (MLE) and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Assuming that the common scale parameter is known, the MLE, uniformly minimum variance unbiased estimator, Bayes estimation and confidence interval of R are obtained. The MLE of R, asymptotic distribution of R in the general case, is also discussed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.     

Estimation of P(Y < X) for progressively first-failure-censored generalized inverted exponential distribution

In this article, we consider the problem of estimation of the stress–strength parameter δ?=?P(Y?<?X) based on progressively first-failure-censored samples, when X and Y both follow two-parameter generalized inverted exponential distribution with different and unknown shape and scale parameters. The maximum likelihood estimator of δ and its asymptotic confidence interval based on observed Fisher information are constructed. Two parametric bootstrap boot-p and boot-t confidence intervals are proposed. We also apply Markov Chain Monte Carlo techniques to carry out Bayes estimation procedures. Bayes estimate under squared error loss function and the HPD credible interval of δ are obtained using informative and non-informative priors. A Monte Carlo simulation study is carried out for comparing the proposed methods of estimation. Finally, the methods developed are illustrated with a couple of real data examples.     

Robust inference for the stress–strength reliability

We address the problem of robust inference about the stress–strength reliability parameter R = P(X < Y), where X and Y are taken to be independent random variables. Indeed, although classical likelihood based procedures for inference on R are available, it is well-known that they can be badly affected by mild departures from model assumptions, regarding both stress and strength data. The proposed robust method relies on the theory of bounded influence M-estimators. We obtain large-sample test statistics with the standard asymptotic distribution by means of delta-method asymptotics. The finite sample behavior of these tests is investigated by some numerical studies, when both X and Y are independent exponential or normal random variables. An illustrative application in a regression setting is also discussed.     

Estimation of the Reliability of a Stress-Strength System from Power Lindley Distributions

In this paper, we are interested in the estimation of the reliability parameter R = P(X > Y) where X, a component strength, and Y, a component stress, are independent power Lindley random variables. The point and interval estimation of R, based on maximum likelihood, nonparametric and parametric bootstrap methods, are developed. The performance of the point estimate and confidence interval of R under the considered estimation methods is studied through extensive simulation. A numerical example, based on a real data, is presented to illustrate the proposed procedure.     

Estimation of reliability of a component subjected to bivariate exponential stress

In this paper, we estimate the reliability of a component subjected to two different stresses which are independent of the strength of a component. We assume that the distribution of stresses follow a bivariate exponential (BVE) distribution. If X is the strength of a component subjected to two stresses (Y ₁,Y ₂), then the reliability of a component is given by R=P[Y ₁+Y ₂<X]. We estimate R when (Y ₁,Y ₂) follow different BVE models proposed by Marshall-Olkin (1967), Block-Basu-(1974), Freund (1961) and Proschan-Sullo (1974). The distribution of X is assumed to be exponential. The asymptotic normal (AN) distributions of these estimates of R are obtained.     

Estimation of P{X < Y} for geometric—exponential model based on complete and censored samples

This article deals with the estimation of R = P{X < Y}, where X and Y are independent random variables from geometric and exponential distribution, respectively. For complete samples, the MLE of R, its asymptotic distribution, and confidence interval based on it are obtained. The procedure for deriving bootstrap-p confidence interval is presented. The UMVUE of R and UMVUE of its variance are derived. The Bayes estimator of R is investigated and its Lindley's approximation is obtained. A simulation study is performed in order to compare these estimators. Finally, all point estimators for right censored sample from the exponential distribution, are obtained.     

Estimation of reliability from Marshall–Olkin extended Lomax distributions

In this paper, we are mainly interested in estimating the reliability R=P(X>Y) in the Marshall–Olkin extended Lomax distribution, recently proposed by Ghitany et al. [Marshall–Olkin extended Lomax distribution and its application, Commun. Statist. Theory Methods 36 (2007), pp. 1855–1866]. The model arises as a proportional odds model where the covariate effect is replaced by an additional parameter. Maximum likelihood estimators of the parameters are developed and an asymptotic confidence interval for R is obtained. Extensive simulation studies are carried out to investigate the performance of these intervals. Using real data we illustrate the procedure.     

Note on estimation of reliability under bivariate pareto stress-strength model

In this paper, we discuss the problem of estimating reliability (R) of a component based on maximum likelihood estimators (MLEs). The reliability of a component is given byR=P[Y<X]. Here X is a random strength of a component subjected to a random stress(Y) and (X,Y) follow a bivariate pareto(BVP) distribution. We obtain an asymptotic normal(AN) distribution of MLE of the reliability(R).     

The simplicity of likelihood based inferences for P(X < Y) and for the ratio of means in the exponential model

The profile likelihood of the reliability parameter θ = P(X < Y) or of the ratio of means, when X and Y are independent exponential random variables, has a simple analytical expression and is a powerful tool for making inferences. Inferences about θ can be given in terms of likelihood-confidence intervals with a simple algebraic structure even for small and unequal samples. The case of right censored data can also be handled in a simple way. This is in marked contrast with the complicated expressions that depend on cumbersome numerical calculations of multidimensional integrals required to obtain asymptotic confidence intervals that have been traditionally presented in scientific literature.     

Asymptotic and Resampling-Based Confidence Intervals for P(X < Y)

ABSTRACT

We consider asymptotic and resampling-based interval estimation procedures for the stress-strength reliability P(X < Y). We developed and studied several types of intervals. Their performances are investigated using simulation techniques and compared in terms of attainment of the nominal confidence level, symmetry of lower and upper error rates, and expected length. Recommendations concerning their use are given.     

Weighted likelihood based inference for P(X < Y)

This contribution deals with the statistical problem of evaluating the stress–strength reliability parameter R = P(X < Y), when both stress and strength data are prone to contamination. Standard likelihood inference can be badly affected by mild data inadequacies, that often occur in the form of several outliers. Then, robust tools are recommended. Here, inference relies on the weighted likelihood methodology. This approach has the advantage to lead to robust estimators, tests, and confidence intervals that share the main asymptotic properties of their classical counterparts. The accuracy of the proposed methodology is illustrated both by numerical studies and real-data applications.     

Estimation of reliability from a bivariate log-normal data

It has been established that the bivariate log-normal distribution is appropriate for modelling certain paired observations. In this paper, we have developed large-sample confidence intervals of the dependence and reliability R=P(X>Y) parameters from a bivariate log-normal distribution with equal log-normal means. The parameter R provides a general measure of difference between the two populations and has applications in many areas. The performance of these confidence intervals has been examined by extensive simulation studies. The results are illustrated with an example dealing with a quantitative assay problem.