On the Maximum Likelihood Method for Censored Bivariate Samples

Abstract

The result presented in this note concerns the limiting behavior of maximum likelihood estimators for type II censored samples from a bivariate absolutely continuous distribution. Let (X₁,Y₁),…, (X_n,Y_n) be independent and identically distributed (iid) bivariate random variables (r.v’s) from the parent population. One assumes that only r (r < n and r/n → p ∈ (0,1), n→∞) pairs of (X,Y)’s are observed in which r smallest order statistics of X’s occur. In these pairs suitable Y-covariates are known as induced order statistics or concomitants of order statistics. Sufficient conditions for the strong consistency and asymptotic normality of maximum likelihood estimators of a vector-valued parameter are established. The treatment is similar as in the classic iid case. Obtained results follow the asymptotics of functions of order statistics and some properties of the induced order statistics.