kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno's Formula

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 Faà di Bruno's formula, which therefore has been 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

Modelling the Frequency of Interarrival Times and Rainfall Depths with the Poisson Hurwitz-Lerch Zeta Distribution

“…Using integer partitions, the explicit expression of the partial exponential polynomials can be recovered in R using the kStatistics package [16]. A useful property used in the following is…”

“…Thanks to this device, the complete exponential Bell polynomials results as a special case of a wider class of polynomial families, the generalized partition polynomials [16] B n (a 1 ,. .…”

“…Proof. Using the binomial expansion of y k = (y − a + a) k , we have (16) follows by taking into account (14).…”

“…where {ξ j } are the moments of Y d = HLZD(a,θ,s), given in (16), and L n is the n-th logarithmic polynomial (12). Let us recall that, if the moment generating function (mgf) M Y (t) of Y is well defined in a suitable neighborhood of 0, then the coefficients {κ n (Y)} in the expansion…”

On the Cumulants of the First Passage Time of the Inhomogeneous Geometric Brownian Motion

“…The second one is that this case is simpler from a numerical point of view and the main scope of the present paper is to propose a mathematical approach rather than perform a complete study of the computational properties of the numerical routine. However, we stress that cumulants of any order can be evaluated numerically by implementing a symbolic-numeric procedure using the R package kStatistics [41]. Indeed the computation of the k-th formal cumulant c * k in (30) involves the computation of the k-th logarithmic polynomial (13), which is a special case of the general partition polynomial (16).…”