On the Nile Problem by Sir Ronald Fisher

TL;DR

This paper introduces a new purely statistical method to analyze the Nile problem, demonstrating that no unbiased estimator with minimal variance exists under mild conditions, and extends to curved exponential families.

Contribution

A novel statistical approach is developed to study UMVUEs in curved exponential families, addressing longstanding questions about the Nile problem.

Findings

Proves no UMVUE exists for the Nile problem under mild conditions.

Develops a method applicable to any family with an ancillary statistic.

Extends analysis to curved exponential families with polynomial constraints.

Abstract

The Nile problem by Ronald Fisher may be interpreted as the problem of making statistical inference for a special curved exponential family when the minimal sufficient statistic is incomplete. The problem itself and its versions for general curved exponential families pose a mathematical-statistical challenge: studying the subalgebras of ancillary statistics within the $\sigma$-algebra of the (incomplete) minimal sufficient statistics and closely related questions of the structure of UMVUEs. In this paper a new method is developed that, in particular, proves that in the classical Nile problem no statistic subject to mild natural conditions is a UMVUE. The method almost solves an old problem of the existence of UMVUEs. The method is purely statistical (vs. analytical) and works for any family possessing an ancillary statistic. It complements an analytical method that uses only the first order ancillarity (and thus works when the existence of ancillary subalgebras is an open problem) and works for curved exponential families with polynomial constraints on the canonical parameters of which the Nile problem is a special case.