Some large-concentration-parameter asymptotics for the k-class estimators

Abstract

A sufficient condition is derived in this paper for the consistency and asymptotic normality of the k-class estimators ( k-stochastic or nonstochastic) as the concentration parameter increases indefinitely, with the sample size either staying fixed or also increasing. It is further shown that the limited-information maximum likelihood estimator satisfies this condition. Since large sample size implies a large concentration parameter, but not vice versa, the usual conditions for consistency and asymptotic normality of the k-class estimators as the sample size increases can be inferred from the results given in this paper. But more importantly, the results in this paper shed further light on the small-sample properties of the stochastic k-class estimators and can serve as a starting point for the derivation of asymptotic approximations for these estimators as the concentration parameter goes to infinity, while the sample size either stays fixed or also goes to infinity.