Asymptotic Results for Inference Procedures Based on the $r$ Smallest Observations

Abstract

We consider procedures for statistical inference based on the smallest $r$ observations from a random sample. This method of sampling is of importance in life testing. Under weak regularity conditions which include the existence of a q.m. derivative for the square root of the ratio of densities, we obtain an approximation to the likelihood and establish the asymptotic normality of the approximation. This enables us to reach several important conclusions concerning the asymptotic properties of point estimators and of tests of hypotheses which follow directly from recent developments in large sample theory. We also give a result for expected values which has importance in the theory of rank tests for censored data.