Annals of Statistics

Large Deviation Probabilities for Certain Nonparametric Maximum Likelihood Estimators

J. Pfanzagl

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 18, Number 4 (1990), 1868-1877.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347884

Digital Object Identifier
doi:10.1214/aos/1176347884

Mathematical Reviews number (MathSciNet)
MR1074441

Zentralblatt MATH identifier
0721.62048

JSTOR
links.jstor.org

Subjects
Primary: 62F10: Point estimation
Secondary: 62F12: Asymptotic properties of estimators

Keywords
Maximum likelihood estimators consistency nonparametric models mixtures

Citation

Pfanzagl, J. Large Deviation Probabilities for Certain Nonparametric Maximum Likelihood Estimators. Ann. Statist. 18 (1990), no. 4, 1868--1877. doi:10.1214/aos/1176347884. https://projecteuclid.org/euclid.aos/1176347884


Export citation