A new method for fast computing unbiased estimators of cumulants

TL;DR

This paper introduces a symbolic umbral calculus-based method for rapidly computing unbiased estimators of cumulants, significantly improving computational efficiency over existing algorithms.

Contribution

It presents a novel, fast algorithm for generating k-statistics and polykays using umbral calculus, extending to multivariate cases.

Findings

Algorithms are significantly faster than existing methods.

The method effectively handles multivariate cumulant estimations.

Connection between cumulants and compound Poisson variables is key.

Abstract

We propose new algorithms for generating $k$-statistics, multivariate $k$-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up is obtained by means of a symbolic method arising from the classical umbral calculus. The classical umbral calculus is a light syntax that involves only elementary rules to managing sequences of numbers or polynomials. The cornerstone of the procedures here introduced is the connection between cumulants of a random variable and a suitable compound Poisson random variable. Such a connection holds also for multivariate random variables.