Quick Consistency of Quasi Maximum Likelihood Estimators

Abstract

A family of probability measures $\mathscr{P}$ on some measurable space $(X, \mathscr{A})$ and a class of estimator sequences $\hat{P}_n: X^n \rightarrow \mathscr{P}, n \in \mathbb{N}$, containing maximum likelihood estimators are considered. For $P \in \mathscr{P}$ it is proved that there are numbers $c > 0, h_0 > 0$ fulfilling $P^n\{n^{1/2} d(\hat{P}_n, P) > h\} \leq \exp(-ch^2)$ for $n \in \mathbb{N}, h \geq h_0$, where $d$ denotes the Hellinger distance of probability measures. Then parameterized families $\mathscr{P} = \{P(\theta): \theta \in \Theta\}$ are considered where $(\Theta, \Delta)$ is a separable and finite-dimensional metric space, and for sequences $\hat{\Theta}_n: X^n \rightarrow \Theta, n \in \mathscr{N}$, estimating the parameter similar inequalities are derived.