Some incomplete and boundedly complete families of discrete distributions

TL;DR

This paper introduces a general framework for identifying incomplete and boundedly complete families of discrete distributions, characterizing unbiased estimators, and providing numerous new examples of such families.

Contribution

It presents a unified result that constructs and characterizes incomplete and boundedly complete discrete distribution families, expanding the known examples and theoretical understanding.

Findings

Explicit characterization of unbiased estimators of zero

Construction of many new incomplete and boundedly complete families

Identification of uniformly minimum variance unbiased estimators

Abstract

We present a general result giving us families of incomplete and boundedly complete families of discrete distributions. For such families, the classes of unbiased estimators of zero with finite variance and of parametric functions which will have uniformly minimum variance unbiased estimators with finite variance are explicitly characterized. The general result allows us to construct a large number of families of incomplete and boundedly complete families of discrete distributions. Several new examples of such families are described.