A New Nonparametric Estimator of the Center of a Symmetric Distribution

Schuster, E. F. ; Narvarte, J. A.

Ann. Statist., Tome 1 (1973) no. 2, p. 1096-1104 / Harvested from Project Euclid

Résumé

Let $F_n(x)$ be the empirical distribution function based on a random sample of size $n$ from a continuous symmetric distribution with center $\theta$. As a nonparametric estimator of $\theta$, we propose $a^\ast$ where $a^\ast$ is chosen so as to minimize the function $h$ where $h(a) = \max_x |F_n(x) + F_n((2a - x)^-) - 1|$. In this paper we present an algorithm for constructing the interval of all $a$ which minimize $h$. We show that if $a^\ast$ is chosen as the center of this interval then $a^\ast$ is an unbiased estimator of $\theta$ which converges to $\theta$ with probability one at a rate of $n^{1/2-\delta}$ for $\delta > 0$. We then use the large or small sample distribution of $h(\theta)$ given by Butler (1969) to construct confidence intervals for $\theta$ and show how one can test for symmetry when the center is not specified under the null hypothesis.

Détail

Publié le : 1973-11-14
Classification:  Nonparametric estimator,  center of symmetry,  symmetric distribution,  empirical distribution function,  62G05