The Construction of Uniformly Minimum Variance Unbiased Estimators for Exponential Distributions

Abstract

Consider a sample $(x_1, x_2, \cdots, x_N)$ from a population with a distribution function $F_\theta(x), (\theta \epsilon \mathbf{\Omega})$ for which a complete sufficient statistic, $s(x)$, exists. Then any parametric function $g(\theta)$ possesses a unique minimum variance unbiased estimator U.M.V.U.E., which may be obtained by the Rao-Blackwell theorem provided an unbiased estimator of $g(\theta)$ with finite variance for each $\theta \epsilon \mathbf{\Omega}$ is available. In this paper we will consider the Koopman-Darmois class of exponential densities and develop a method for obtaining the U.M.V.U.E., $t_g$, of $g(\theta)$ without explicit knowledge of any unbiased estimator of $g(\theta)$. The U.M.V.U.E. $t_g$ is given as the limit in the mean (l.i.m.) of a series and a convergent series is also given for the variance. For any arbitrary but fixed $\theta_0 \epsilon \mathbf{\Omega}$, it can be verified that the complete sufficient statistic $s(x)$ has moments of all orders and that these moments determine its distribution function. Hence the set of polynomials in $s(x)$ is dense in the Hilbert space, $V$ (with the usual inner product), of Borel measurable functions of $s(x)$. Since $t_g$ is an element of $V$, we may obtain a generalized Fourier series for it by constructing a complete orthonormal set $\{\varphi_n\}$ for $V$. Such a set $\{\varphi_n\}$ may be obtained from the density function and its derivatives with respect to $\theta$. For a subclass of the exponential family, Seth [18] has obtained $\{\varphi_n\}$ in a form which is convenient for our purposes. We will study this case in Section 3 and use Seth's results to give an explicit construction of $t_g$. Criteria for the pointwise convergence of the series will also be given. In Section 4 examples illustrating the use of the method are given and some related results are discussed. The general theory for the representation of minimum variance unbiased estimates, both local and uniform, has been developed in depth, for example in [5], [18], [19], [16], [3], and [4]. The present remarks, though founded in the general theory (in particular [3] and [4]), are tailored specifically to the exponential family.