Through the classical umbral calculus, we provide a unifying syntax for single and multivariate kstatistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for k-statistics that results in a very fast algorithm to generate such estimators. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 2, 440-468. This reprint differs from the original in pagination and typographic detail. 1350-7265

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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.

By means of the notion of umbrae indexed by multisets, a general method to express estimators and their products in terms of power sums is derived. A connection between the notion of multiset and integer partition leads immediately to a way to speed up the procedures. Comparisons of computational times with known procedures show how this approach turns out to be more efficient in eliminating much unnecessary computation.

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