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

TL;DR

This paper introduces an inversion principle for unbiased estimation of products of moments and cumulants in finite and infinite populations, providing explicit formulas up to order 6 and identifying key functionals with specific eigenvalues.

Contribution

It establishes a novel inversion principle for unbiased estimation of moments and cumulants, with explicit formulas and characterization of functionals with eigenvalue properties.

Findings

Proves the inversion principle for unbiased estimates.

Provides explicit formulas for moments and cumulants up to order 6.

Identifies functionals with specific eigenvalues as noncentral moments and certain central moments.

Abstract

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)$ be any product of the moments or cumulants of $F_N$. Let $T_{n, N} \left( F_N \right) = E T \left( F_n \right)$. 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 $E T \left( F_n \right) = \lambda_{n, N} T \left( F_N \right)$ 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.