Uniformly Minimum Variance Estimation in Location Parameter Families

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 $\operatorname{df}$.