kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Fa\`a Di Bruno's Formula

TL;DR

kStatistics is an R package that provides unbiased estimators for univariate and multivariate cumulants and their products, utilizing an algorithm based on multivariate Faà di Bruno's formula for efficient computation and polynomial family generation.

Contribution

The paper introduces a unified R package, kStatistics, that implements an algorithm for unbiased cumulant estimation and incorporates multivariate Faà di Bruno's formula for versatile polynomial generation.

Findings

Provides unbiased cumulant estimators with minimum variance.

Includes implementation of multivariate Faà di Bruno's formula.

Enables generation of various polynomial families, such as Bell polynomials.

Abstract

kStatistics is a package in R that serves as a unified framework for estimating univariate and multivariate cumulants as well as products of univariate and multivariate cumulants of a random sample, using unbiased estimators with minimum variance. The main computational machinery of kStatistics is an algorithm for computing multi-index partitions. The same algorithm underlies the general-purpose multivariate Fa\`a di Bruno's formula, which has been therefore included in the last release of the package. This formula gives the coefficients of formal power series compositions as well as the partial derivatives of multivariable function compositions. One of the most significant applications of this formula is the possibility to generate many well-known polynomial families as special cases. So, in the package, there are special functions for generating very popular polynomial families, such as the Bell polynomials. However further families can be obtained, for suitable choices of the formal power series involved in the composition or when suitable symbolic strategies are employed. In both cases, we give examples on how to modify the R codes of the package to accomplish this task. Future developments are addressed at the end of the paper.