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May, 1975 Uniformly Minimum Variance Estimation in Location Parameter Families
Lennart Bondesson
Ann. Statist. 3(3): 637-660 (May, 1975). DOI: 10.1214/aos/1176343127

Abstract

Let $x_1, \cdots, x_n$ be a sample of size $n$ of an $\operatorname{rv}$ with $\operatorname{df} F(x - \theta)$, where $F$ is known but $\theta$ unknown. In this paper we make a Fourier approach to the problem of existence of a statistic $g(x_1, \cdots, x_n)$ which is a uniformly minimum variance (UMV) estimator of its own mean value. We mention only some of the results. If $n = 1$ we find an NASC for an estimator $g(x_1)$ to be, in a restricted sense, UMV. This condition is given in terms of the zeroes of the ch.f. of $F$ and the support of the Fourier transform of $g$. If $n \geqq 2$, it is shown that a statistic of the form $g(\bar{x})$, where $\bar{x}$ is the sample mean, cannot be UMV, unless $g$ is periodic or $F$ is a normal $\operatorname{df}$. We prove the non-existence of a UMV-estimator of $\theta$, provided that the tail of $F$ tends to zero rapidly enough. Finally, it is proved that no polynomial $P(x_1, \cdots, x_n)$ can be a UMV-estimator, unless $F$ is a normal $\operatorname{df}$.

Citation

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Lennart Bondesson. "Uniformly Minimum Variance Estimation in Location Parameter Families." Ann. Statist. 3 (3) 637 - 660, May, 1975. https://doi.org/10.1214/aos/1176343127

Information

Published: May, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0303.62024
MathSciNet: MR652532
Digital Object Identifier: 10.1214/aos/1176343127

Subjects:
Primary: 62F10
Secondary: 62E10

Keywords: Cauchy's functional equation , Characteristic function , convolution , entire analytic function of finite order , Fourier transform , generalized function , location parameter , Pitman estimator , spectrum of a function , unbiased estimator of zero , uniformly minimum variance estimator

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 3 • May, 1975
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