Multivariate stochastic comparisons of mixture models

Let $$X_1,\ldots ,X_n$$ X 1 , … , X n be a random sample from a distribution function $$F$$ F that denote lifetimes of $$n$$ n components of a coherent system. Suppose the system fails at $$X_{k:n}$$ X k : n , the $$k$$ k th order statistic of $$X$$ X ’s, since we are not aware of the exact time at which the system has been failed, the residual lifetimes of the remaining $$n-k$$ n - k components, denoted by $$X^{(k)}_1,\ldots ,X^{(k)}_{n-k}$$ X 1 ( k ) , … , X n - k ( k ) , are no longer independent but exchangeable. In this paper, multivariate stochastic comparisons of two vectors of lifetimes of the remaining components in the two sample problems are studied. Some sufficient conditions under which multivariate mixture models are compared stochastically with respect to the multivariate likelihood ratio ordering, the multivariate hazard rate ordering and the multivariate reversed hazard rate ordering are provided. These comparisons are done for different choices of the mixed distributions as well as mixing distributions. The new results obtained are applied to compare multivariate mixtures of location models. Copyright Springer-Verlag Berlin Heidelberg 2015