Unbiased Estimation: Functions of Location and Scale Parameters

Abstract

Unbiased estimators for functions of a location parameter $\theta$ and a scale parameter $\rho$ are expressed as unknown functions in integral equations of convolution type, and are then obtained by integral transform methods. An outline of the paper is contained in Section 3. The main results consist in the application of various derived expressions to the exponential distribution with parameters $\theta$ and $\rho$, the gamma and Weibull distributions with parameter $\rho$, and to general distributions with truncation parameter $\theta$. In the latter case, a simple formula is given for a minimum variance unbiased estimator of any absolutely continuous function of $\theta$; this extends slightly a result of Davis [3] concerning distributions of exponential type. Throughout the paper particular attention is paid to the estimation of the probability that a single observation will lie in a certain Borel set, when this probability is regarded as a function of the parameters $\theta$ and/or $\rho$. Extensions to sample points of $m$ observations and Borel sets in $m$-space are in most cases immediate.