Asymptotic properties of a solution to the likelihood equation with life-testing applications

Abstract

The asymptotic properties of a solution of the maximum likelihood equation for the case of independent and nonidentically distributed random variables are considered. A set of sufficient conditions for its consistency and asymptotic normality is given. As an application, suppose Y_i(i = 1, 2, ...., n) has exponential density f_θ(y) = θ ε^y, θ > 0, y > 0, and is censored if Y_i > c_i. Let X_i = min (Y_i, c_i) be the censored observation. A sufficient condition for the asymptotic normality of a solution of the likelihood equation is obtained as a special case of the general theorem. Justification for the Edgeworth expansion for this estimate is provided for a special case. We also consider the problem of estimating θ when instead of X_i one can observe only whether Y_i has exceeded c_i.