Partially complete sufficient statistics are jointly complete

TL;DR

This paper advances the theory of statistical completeness by proving a new theorem about partially complete sufficient statistics, with two concise proofs and illustrative examples, enriching the foundational understanding of statistical sufficiency.

Contribution

It establishes a previously overlooked theorem linking partial and joint completeness of sufficient statistics, with two novel proofs and discussion of related results.

Findings

Theorem on partial and joint completeness proved.

Two concise proofs provided for the main result.

Illustrative examples demonstrate the theorem's applications.

Abstract

The theory of the basic statistical concept of (Lehmann-Scheff\'e-)completeness is perfected by providing the theorem indicated in the title and previously overlooked for several decades. Relations to earlier results are discussed and illustrating examples are presented. Of the two proofs offered for the main result, the first is direct and short, following the prototypical example of Landers and Rogge (1976), and the second is very short and purely statistical, utilizing the basic theory of optimal unbiased estimation in the little known version completed by Schmetterer and Strasser (1974).