On the Nile Problem by Sir Ronald Fisher

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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 the paper a new method is developed that proves that in the classical Nile problem no statistic subject to mild natural conditions is a UMVUE. The result almost solves an old problem of the existence of UMVUEs. The method is purely statistical (vs. analytical) and requires the existence of an ancillary subalgebra. An analytical method that uses only first order ancillarity (and thus works in the setups when the existence of ancillary subalgebras is an open problem) proves nonexistence of UMVUEs for curved exponential families with polynomial constraints on the parameters.

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