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 $\hat{F}=F_N$ be the empirical distribution of a sample of size $n$ drawn from the population without replacement. We prove the following remarkable {\it inversion principle} for obtaining unbiased estimates. Let $T\left(F_N\right)$ be any product of the moments or cumulants of $F_N$. Let $T_{n,N}\left(F_N\right)=ET\left(F_n\right)$. Then $E{T}_{N,n}\left(F_n\right)=T\left(F_N\right)$. We also obtain an explicit expression for $T_{n,N}\left(F_N\right)$ for all $T\left(F_N\right)$ 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 $ET\left(F_n\right)={\lambda }_{n,N}T\left(F_N\right)$ are noncentral moments, and generalized second and third order central moments. For these three cases the eigenvalues are ${\lambda }_{n,N}=1$, $\left(1-n^{-1}\right){\left(1-N^{-1}\right)}^{-1}$, and $\left(1-n^{-1}\right)\left(1-2n^{-1}\right){\left(1-N^{-1}\right)}^{-1}{\left(1-2N^{-1}\right)}^{-1}$ respectively.