Unbiased estimates for products of moments and cumulants for finite and infinite populations

Abstract

Let F = F N be the distribution of a finite real population of size N . Let F = F N be the empirical distribution of a sample of size n drawn from the population without replacement. We prove the following remarkable inversion principle for obtaining unbiased estimates. Let T (F N ) be any product of the moments or cumulants ofWe also obtain an explicit expression for T n,N (F N ) for all T (F N ) of order up to 6.We also prove the following related result. If F n and F N are the sample and population distributions, the only functionals for which ET (F n ) = λ n,N T (F N ) are noncentral moments, and generalized second and third order central moments. For these three cases the eigenvalues are λ n,N =