On the combinatorics of k-statistics

Di Nardo elvira, University of Turin

Quite often numerical computing helps in searching for solutions with a required level of precision, where no closed-form solutions are available. Unlike numerical methods that deal with data, symbolic methods handle objects that are either algebraic or formal expressions. When used properly, they can give us more insight into the problem we are trying to solve, since symbolic methods might yield closed or explicit formulas. While the expression "symbolic method" is also used in different contexts, here we refer to a collection of manipulation techniques meant to carry out algebraic calculations, preferably employing an algorithmic approach, in order to discover efficient mechanical processes that can be passed to a computer. Usually, this is referred to as "symbolic computation." One instance that bears particular significance is the computation of k-statistics, a challenging problem since their first introduction by Fisher as unbiased estimators of cumulants. To accomplish this computation in a reasonable amount of time, numerous authors have proposed various methods. A suitable generalization of randomized compound Poisson random variables can be used to achieve an efficient computation of k-statistics by employing the symbolic method, arising from the so-called "classical umbral calculus". The underlying combinatorial devices have been extended to polykays, unbiased estimators of cumulant products, as well as the multivariate case. Recently, all these procedures have been merged into a R package. The main computational machinery 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. One of the most significant applications of this formula is the possibility to generate many well-known polynomial families as special cases.

Area: CS1 - Algebraic methods in Statistics and Probability (Elvira Di Nardo)

Keywords: k-statistics, multivariate Faà di Bruno’s formula, umbral calculus

Please Login in order to download this file