## Electronic Journal of Statistics

### On the Nile problem by Sir Ronald Fisher

#### 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.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 1968-1982.

Dates
First available in Project Euclid: 18 July 2013

https://projecteuclid.org/euclid.ejs/1374153370

Digital Object Identifier
doi:10.1214/13-EJS832

Mathematical Reviews number (MathSciNet)
MR3084678

Zentralblatt MATH identifier
1293.62009

Subjects
Primary: 62B05: Sufficient statistics and fields
Secondary: 62F10: Point estimation

#### Citation

Kagan, Abram M.; Malinovsky, Yaakov. On the Nile problem by Sir Ronald Fisher. Electron. J. Statist. 7 (2013), 1968--1982. doi:10.1214/13-EJS832. https://projecteuclid.org/euclid.ejs/1374153370

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