The Annals of Statistics

Adaptive Maximum Likelihood Estimators of a Location Parameter

Charles J. Stone

Abstract

Consider estimators $\hat{\theta}_n$ of the location parameter $\theta$ based on a sample of size $n$ from $\theta + X$, where the random variable $X$ has an unknown distribution $F$ which is symmetric about the origin but otherwise arbitrary. Let $\mathscr{F}$ denote the Fisher information on $\theta$ contained in $\theta + X$. We show that there is a nonrandomized translation and scale invariant adaptive maximum likelihood estimator $\hat{\theta}_n$ of $\theta$ which doe not depend on $F$ such that $\mathscr{L}(n^{\frac{1}{2}}(\hat{\theta}_n - \theta)) \rightarrow N(0, 1/\mathscr{J})$ as $n \rightarrow \infty$ for all symmetric $F$.

Article information

Source
Ann. Statist. Volume 3, Number 2 (1975), 267-284.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176343056

Digital Object Identifier
doi:10.1214/aos/1176343056

Mathematical Reviews number (MathSciNet)
MR362669

Zentralblatt MATH identifier
0303.62026

JSTOR