Estimation And Properties Of U Statistics Of Reliability For Complex System

This paper studies some properties of complex system reliability estimation and partial properties of complex system reliability U statistics estimation, according to the description of complex system on the basis of the Minimal cut sets and the minimum path, which is divided into the following three parts.In the first part, based on the analytical expression of reliability for the complex system which is described by minimal cut sets, the expectation and variance of complex system reliability estimation are given. When the minimal cut sets have some particular features, the variance of the estimation for the complex system reliability is easly calculated.In the second part, this paper will introduce the definition of m sample U statistics by the definitions of one sample U statistics and two sample U statistics. Though the method for calculating expectation and variance of one sample and two sample u statistics, the expectation and variance of m sample U statistics will be given and the result will be applied to U statistics for estimating complex system reliability. By constructing special operations of vectors and matrix, we obtain variance of the U statistics for estimating complex system reliability.In the third part, the asymptotic normality of m sample U statistics is studied, on the basis of obtained expectation and variance of m sample U statistics. By constructing special random variable set, calculating the function of statistics composed by m sample U statistics on the set, and then applying Slutsky Theorem, finally, the asymptotic normality of complex system reliability U statistics is proved, with the combination of the obtained expression of complex system reliability U statistics.