Strong Consistency of Approximate Maximum Likelihood Estimators with Applications in Nonparametrics

Abstract

Wald's general analytic conditions that imply strong consistency of the approximate maximum likelihood estimators (AMLEs) have been extended by Le Cam, Kiefer and Wolfowitz, Huber, Bahadur, and Perlman. All these conditions use the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_0)\rbrack$, where $\theta_0$ is the true value of the parameter. However these methods usually fail in the nonparametric case. Thus, in this paper, for each $\theta \neq \theta_0$, we look at the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_r(\theta))\rbrack$, where $\theta_r(\theta)$ is a certain parameter selected in a neighborhood $V_r$ of $\theta_0$. Some general analytic conditions that imply strong consistency of the AMLE are given. The results are shown to be applicable to several nonparametric families having densities, e.g., concave distributions functions, and increasing failure rate distributions. In particular, they can be applied to several censored data cases.