A Tutorial on Multivariate k-Statistics and their Computation
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This document aims to provide an accessible tutorial on the unbiased estimation of multivariate cumulants, using k-statistics. We offer an explicit and general formula for multivariate k-statistics of arbitrary order. We also prove that the k-statistics are unbiased, using Möbius inversion and rudimentary combinatorics. Many detailed examples are considered throughout the paper. We conclude with a discussion of k-statistics computation, including the challenge of time complexity, and we examine a couple of possible avenues to improve the efficiency of this computation. The purpose of this document is threefold: to provide a clear introduction to k-statistics without relying on specialized tools like the umbral calculus; to construct an explicit formula for k-statistics that might facilitate future approximations and faster algorithms; and to serve as a companion paper to our Python library PyMoments, which implements this formula.
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