A New Nonparametric Estimator of the Center of a Symmetric Distribution

Abstract

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.