## The Annals of Statistics

- Ann. Statist.
- Volume 13, Number 3 (1985), 1097-1113.

### Estimation of a Symmetric Distribution

#### Abstract

Suppose that $F_0$ is a population which is symmetric about zero, so that $F(\cdot) = F_0(\cdot - \theta)$ is symmetric about $\theta$. We consider the problem of estimating $F_0$ (shape parameter), both $\theta$ and $F_0$, and $F$ based on a random sample from $F$. First, some asymptotically minimax bounds are obtained. Then, some estimates are constructed which are asymptotically minimax-efficient (the risks of which achieve the minimax bounds uniformly). Furthermore, it is pointed out that one can estimate $F_0$, the shape of $F$, as well without knowing the location parameter $\theta$ as with knowing it. After a slight modification, Stone's (1975) estimator is proved to be asymptotically minimax-efficient in the Hellinger neighborhood.

#### Article information

**Source**

Ann. Statist., Volume 13, Number 3 (1985), 1097-1113.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349658

**Digital Object Identifier**

doi:10.1214/aos/1176349658

**Mathematical Reviews number (MathSciNet)**

MR803760

**Zentralblatt MATH identifier**

0594.62015

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

**Keywords**

Gaussian experiments distribution functions asymptotically minimax estimators location parameters

#### Citation

Lo, Shaw-Hwa. Estimation of a Symmetric Distribution. Ann. Statist. 13 (1985), no. 3, 1097--1113. doi:10.1214/aos/1176349658. https://projecteuclid.org/euclid.aos/1176349658