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November, 1973 A New Nonparametric Estimator of the Center of a Symmetric Distribution
E. F. Schuster, J. A. Narvarte
Ann. Statist. 1(6): 1096-1104 (November, 1973). DOI: 10.1214/aos/1176342559


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.


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E. F. Schuster. J. A. Narvarte. "A New Nonparametric Estimator of the Center of a Symmetric Distribution." Ann. Statist. 1 (6) 1096 - 1104, November, 1973.


Published: November, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0275.62034
MathSciNet: MR362681
Digital Object Identifier: 10.1214/aos/1176342559

Primary: 62G05

Keywords: center of symmetry , Empirical distribution function , nonparametric estimator , symmetric distribution

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 6 • November, 1973
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