Asymptotic properties of estimators in a binomial reliability growth model and its continuous-time analog

This article deals with a nonhomogeneous binomial (NHB) model where the probability of failure at the ith trial has the functional form μ[iβ - (i) - 1)β], 0 < μ < 1, 0 < β < 1. The model arises in the context of reliability growth of a one-shot system during the successive stages of its development, and is a discrete analog of a continuous-time growth model based on a nonhomogeneous Poisson process (NHPP) with Weibull intensity. Asymptotic properties of the 'continuous analog estimators' (CAE's) are compared with those of the maximum likelihood estimators (MLE's) in the continuous-time growth model. Unlike the NHPP model, the MLE's in the NHB model are not available in closed forms, and derivation of their asymptotic properties confronts the fundamental difficulty that the appropriately scaled information matrix is asymptotically singular. To get around this difficulty, we first link the CAE's to the maximization of a certain pseudo-likelihood, examine the corresponding Taylor expansion, and relate that to the expansion of the correct NHB likelihood. The closeness of the two expansions is then used to establish consistency and asymptotic normality of the MLE's and asymptotic equivalence with the CAE's