Journal of Iranian Statistical Society
Volume:7 Issue: 1, 2008

In this paper, the evaluation of reliability function, Vesely- Fussell measure of component importance and Birnbaum reliability measure of component importance in a consecutive-k-out-of-n:F system and a consecutive-k-out-of-n:G system are considered. Using theminimal cut (path) sets of a consecutive-k-out-of-n:G(F) system, we present nonrecursive algorithms for determining the system reliability and measures of component importance of these systems. We show that these algorithms leads to explicit formulas for determining the reliability function and measures of component importance in a k-out-of-n:F system with independent but not identical components.

We study the problem of testing the hypothesis that the mean vector of a random vector belongs to a given set. For this purpose, we consider a semiparametric mixture of Dirichlet process model in which the mean vector has a prior distribution concentratedon the set of interest. A computational method is given to obtain a sample from the posterior distribution of the mean vector. On the basis of this sample, we can obtain the Bayes estimate and the posterior probability that the hypothesis is true. We give a numerical example to demonstrate the application of this method.

Recently attempts have been made to construct some measures to compute the local dependence between two random variables. Bairamov et al. (2003) introduced a measure of local dependence which is essentially an extension of Galton correlation coefficient. In the present paper, we give an extension of the measure oflocal dependence, given by the cited authors, and study some of its properties. In particular, we show that our measure of local dependence can be applied to measure the dependency between two residual lifetime random variables. In this case, we give an estimator of the proposed measure of local dependency based on the bivariate meanresidual lifetime. The connections between different forms of local dependency measure given in this paper and some other concepts of dependency are also investigated.

This article presents a formula and a series for approximating the normal distribution function. Over the whole range of the normal variable z, the proposed formula has the greatest absolute error less than 6.5e − 09, and series has a very high accuracy. Weexamine the accuracy of our proposed formula and series for various values of z’s. In the sense of accuracy, our formula and series are superior to other formulae and series available in the literature. Based on the proposed formula an extended table for the mean range of the normal variables is established.

In this research, the generalized maximum likelihood estimator (GMLE) is used to investigate the parameters estimation for weighted distributions. There exist situations where the random sample from the population of interest is not available due to the datahaving unequal probabilities of entering the sample. The method of weighted distributions models the certainty of the probabilities of the events as observed and recorded. It is shown that if the mechanism of sample selection is known up to one unknown parameter, the maximum likelihood estimator (MLE) would be unidentifiable when the conjugate weight function is used. This problem is solved by additionof a prior distribution on model parameters yielding the GMLEs which are identifiable. We also propose the GMLEs for negative exponential, normal and Poisson weighted distributions when MLEs are unidentifiable.