Comparisons of multistate models with discrete-time pure-birth process for recurrent events and uncertain parameters

This article concerns a multistate model with recurrent events, which are modeled by a discrete-time pure-birth process. The probability distribution of the lifetime of the multistate system is based on random sums with support on the natural numbers and with geometric summands, whose exact distribution is given in Belzunce et al. (2009) for independent inter-arrival times. We analyze this multistate model by using stochastic orderings, under different statistical assumptions (of independence and dependence of the inter-arrival times and for arbitrarily distributed inter-arrival times) in terms of the comparison of the probabilities that the individual makes another transition having experienced the first k events, for k = 1, 2, …. The effect of the variations and the correlations between the inter-arrival times of the discrete-time pure-birth process on the lifetime of the multistate system is derived. Exact and numerical bounds of the lifetime of the multistate system are described. Also, we provide some qualitative properties of the kind “new better than used” of the lifetime of the multistate system assuming independent and identically distributed summands.