Sign-Preserving Unbiased Estimators in Linear Exponential Families

Abstract

Let ω G∈ Ω be a distribution parameter and r(ω) a positive estimand with range r(Ω). Hoeffding (1984) called an estimator that takes values only in r(Ω), including the boundaries of r(Ω), a range-preserving estimator. The range-preserving property is a sensible requirement to impose on an estimator. If we require unbiasedness, however, such estimators become difficult to find. In this article a less constraining condition, namely sign-preserving unbiasedness, is considered. A sign-preserving unbiased (SPU) estimator for the positive estimand r(ω) is an unbiased estimator taking only nonnegative values. SPU estimators are not necessarily range preserving. Nevertheless, in the case of positive estimands sign preservation is a necessary requirement for range preservation. The class of linear exponential families of order 1 is considered and necessary and sufficient conditions for the existence of SPU estimators are provided. The question of SPU estimability is stated in the language of Laplace transforms, thus making a broad range of powerful mathematical tools available. Illustrations are given for various estimands of common interest, including density functions as a special case.

Key Words:

• Laplace transforms
• Range-preserving estimators
• Uniformly minimum variance unbiased estimators