Fisher's “Problem of the Nile”

Abstract

THE following problem is of central importance in the theory of statistical estimation : k unknown parameters are to be estimated from a sample S consisting of n independent observations from the same parent population; how far can the information in S, relevant to the estimation of the k parameters, be confined to k degrees of freedom? Fisher¹ has shown that a complete solution is possible if a set of k sufficient or of k quasi-sufficient statistics can be found, and it is known (from the work of Segal², further discussed by Bartlett³) that a simultaneous fiducial distribution will then exist, though it may not be complete in the sense of Bartlett. The most general parent population admitting a set of sufficient statistics has been determined by Koopman⁴, but the parallel problem for quasi-sufficiency (Fisher's “Problem of the Nile“) is still unsolved.