Article

# Estimation of P(X <= Y ) for a Bivariate Weibull Distribution

01/2009;
Source: OAI

ABSTRACT

Nous étudions ici l'estimation de R=P(X<Y), quand X et Y sont des variables aleatoires qui suivent la loi bivariate Weibull et X est censurée à Y. On obtient la loi marginale pour les données observées et on en tire MLE,UMVUE,MME de R. Ainsi on obtient les estimateurs de Bayes sur la fonction SEL. On a effectué une simulation de Monte-Carlo pour comparer ces estimateurs.

### Full-text

Available from: Ahmad Parsian, Oct 04, 2015
1 Follower
·
ABSTRACT: In this paper, we consider a system which has $k$ s-independent and identically distributed strength components, and each component is constructed by a pair of s-dependent elements. These elements $(X_{1},Y_{1}),(X_{2},Y_{2}),ldots ,(X_{k},Y_{k})$ follow a Marshall-Olkin bivariate Weibull distribution, and each element is exposed to a common random stress $T$ which follows a Weibull distribution. The system is regarded as operating only if at least $s$ out of $k (1leq sleq k)$ strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}=P$ (at least $s$ of the $(Z_{1},ldots ,Z_{k})$ exceed $T$) where $Z_{i}=min (X_{i},Y_{i})$, $i=1,ldots ,k$. We estimate $R_{s,k}$ by using frequentist and Bayesian approaches. The Bayes estimates of $R_{s,k}$ have been developed by using Lindley's approximation, and the Markov Chain Monte Carlo methods, due to the lack of explicit forms. The asymptotic confidence interval, and the highest probability density credible interval are constructed for $R_{s,k}$. The rel- ability estimators are compared by using the estimated risks through Monte Carlo simulations.