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

Let $F=F_N$ be the distribution of a finite real population of size $N$. Let $\widehat{F}=F_N$ 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)$beanyproductofthemomentsorcumulantsof$F_N$\mathrm{.}Let$T_{n, N} \left( F_N \right) = E T \left( F_n \right)$\mathrm{.}Then$E T_{N, n} \left( F_n \right) = T \left( F_N \right)$\mathrm{.}Wealsoobtainanexplicitexpressionfor$T_{n, N} \left(F_N\right)$forall$ 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 $E T \left( F_n \right) = \lambda_{n, N} T \left( F_N \right)$ 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.