Minimum Variance Unbiased Estimation and Certain Problems of Additive Number Theory

Abstract

Let $T$ be a subset of the set $I$ of non-negative integers. Define $f(\theta) = \sum a(x)\theta^x$ where the summation extends over $T$ and $a(x) > 0, \theta \geqq 0$ with $\theta \epsilon \Theta$, the parameter space, such that $f(\theta)$ is finite and differentiable. One has $\Theta = \{\theta:0 \leqq \theta < R\}$ where $R$ is the radius of convergence of the power series of $f(\theta)$. Then a random variable $X$ with probability function \begin{equation*}\tag{1}\operatorname{Prob}\{X = x\} = p(x, \theta) = a(x)\theta^x/f(\theta) \quad x \epsilon T\end{equation*} is said to have the generalized power series distribution (GPSD) with range $T$ and the series function $f(\theta)$. It may be observed that the range $T$ could be a countable subset of the real numbers. The GPSD is thus an exponential-type discrete distribution; whereas the power series distribution (PSD) as defined by Noack [5], is a special case of a GPSD. The author [7], [8], [9], [10], [11] has discussed some problems of statistical inference associated with the GPSD and some of its particular forms. Roy and Mitra [14] and Guttman [2] have studied the problem of the minimum variance unbiased (MUV) estimation for the PSD's. In this paper, we investigate the problem of the existence of the MVU estimator for the parameter $\theta$ of the GPSD in terms of the number theoretic structure of its range $T$. We further provide the MVU estimators for the probability and distribution functions of the GPSD and consider a few special cases of some practical significance.