## Annals of Statistics

### Large Deviation Probabilities for Certain Nonparametric Maximum Likelihood Estimators

J. Pfanzagl

#### 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

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