A new simulation estimator of system reliability (Q1344651)

Let \(X_ 1,\dots, X_ n\) be Bernoulli random variables such that \[ P(X_ i= i)= \lambda_ i= 1- P(X_ i= 0),\qquad i= 1,\dots, n. \] Let \(W= \sum_{i=1}^ n a_ i X_ i\), where \(a_ i\) are positive constants. Let \(\lambda= \sum_{i= 1}^ na_ i\lambda_ i= EW\). Let \(R\) and \(I\) be random variables defined on the same probability space as \(X_ i\), \(I\) is independent on the vector \(({\mathbf X},{\mathbf R})\) and \[ P(I= i)= a_ i\sum_{i= 1}^ n a_ i,\qquad i= 1,\dots, n. \] The author proves the following equations \[ P(I= i\mid X_ I= 1)= \lambda^{-1}\lambda_ i a_ i, \] \[ E(W R)= \lambda E(R\mid X_ I= 1),\quad P(W> 0)= \lambda E(W^{-1}\mid X_ I= 1). \] Using these identities several new simulation estimators concerning system reliability and multivalue systems are obtained. It is shown that the variance of these estimators is often of the order \(\alpha^ 2\) when the usual raw estimators have variance of the order \(\alpha\) and \(\alpha\) is small. The author also indicates how these estimators can be combined with standard variance reduction techniques of antithetic variables, stratified sampling and importance sampling. The approach can be used, e.g. in the case when one wants via the simulation to estimate \(\alpha\), the probability that an \(m\) component binary system is failed.