## The Annals of Statistics

### Estimation of a Symmetric Distribution

Shaw-Hwa Lo

#### 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

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