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 σ-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.

Partially complete sufficient statistics are jointly complete

“…30-31), first made rigorous by Bahadur (1957) and later more generally by Torgersen (1988) in the mathematically inconvenient and practically less important setting of the UMVU theory, finally led to the present result in the hands of Schmetterer and Strasser (1974), after earlier work of themselves and of Padmanabhan, Linnik, and Rukhin cited by them. Further developments include Bahadur (1976), Kozek (1988), Kagan and Konikov (2006), and Kagan and Malinovsky (2013).…”

On the Structure of UMVUEs

“…For its partial solution see Kagan and Malinovsky (2013). For the relation between sufficient statistic, and sufficient subalgebras we refer to Bahadur (1957).…”