Positive dependence properties of conditionally independent random lifetimes.

Conditions under which conditionally independent random variables are positive dependent are described in this paper. For example, it is shown that if the conditionally independent random variables increase in the hazard rate sense in the condition, then they are WBF (weakened by failures). It is also shown that if the conditionally independent random variables increase in the likelihood ratio sense in the condition, then they are MTP2 (multivariate totally positive of order 2). The results are given for the cases in which the condition is either a random variable or a random vector. Some applications in the areas of imperfect repair, budget allocation, random environments, and discrete-time filtering, illustrate the theory.