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December, 1990 Large Deviation Probabilities for Certain Nonparametric Maximum Likelihood Estimators
J. Pfanzagl
Ann. Statist. 18(4): 1868-1877 (December, 1990). DOI: 10.1214/aos/1176347884


Let $(X, \mathscr{A})$ be a measurable space and $\{P_{\vartheta,\tau}\mid\mathscr{A}: \vartheta \in \Theta, \tau \in T\}$ a family of probability measures. Given an appropriate estimator sequence for $\vartheta$, we define a sequence of asymptotic maximum likelihood estimators for $\tau$ and give bounds for its large deviation probabilities under conditions which are natural for the application to the estimation of mixing distributions. This paper generalizes earlier results of Pfanzagl to the following cases: (i) estimator sequences restricted to a sieve; (ii) estimator sequences using a given estimator sequence for a nuisance parameter; (iii) convergence under the "wrong model;" (iv) large deviation probabilities instead of consistency.


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J. Pfanzagl. "Large Deviation Probabilities for Certain Nonparametric Maximum Likelihood Estimators." Ann. Statist. 18 (4) 1868 - 1877, December, 1990.


Published: December, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0721.62048
MathSciNet: MR1074441
Digital Object Identifier: 10.1214/aos/1176347884

Primary: 62F10
Secondary: 62F12

Keywords: consistency , maximum likelihood estimators , mixtures , nonparametric models

Rights: Copyright © 1990 Institute of Mathematical Statistics

Vol.18 • No. 4 • December, 1990
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