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.