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September, 1982 Quick Consistency of Quasi Maximum Likelihood Estimators
Thomas Pfaff
Ann. Statist. 10(3): 990-1005 (September, 1982). DOI: 10.1214/aos/1176345889


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.


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Thomas Pfaff. "Quick Consistency of Quasi Maximum Likelihood Estimators." Ann. Statist. 10 (3) 990 - 1005, September, 1982.


Published: September, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0499.62033
Digital Object Identifier: 10.1214/aos/1176345889

Primary: 62G05
Secondary: 62F10

Keywords: Consistency of estimators , exponential bounds for consistency , maximum likelihood estimation

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • September, 1982
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