多元正态分布情形下多信息密度的累积量

A new method for fast computing unbiased estimators of cumulants

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 ra…

Multivariate Lag-Windows and Group Representations

Symmetries of the auto-cumulant function (the generalization of the auto-covariance function) of a kth-order stationary time series are derived through a connection with the symmetric group of degree k. Using theory of group representations, symmetries of the auto-cumulant function are demystified and lag-window functions are symmetrized to satisfy these symmetries. A generalized Gabr-Rao optimal kernel, used to estimate general kth-order spectra, is also derived through the developed theory.

Detecting spatial patterns with the cumulant function. Part I: The theory

In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied on this purpose, it can only provide meaningful results if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments quantities and the covariance matrix can only capture the full dependence structure for multivariate Gaussian vectors. Whenever the application at hand can not satisfy t…

Cumulant-free closed-form formulas for some common (dis)similarities between densities of an exponential family

It is well-known that the Bhattacharyya, Hellinger, Kullback-Leibler, $α$-divergences, and Jeffreys' divergences between densities belonging to a same exponential family have generic closed-form formulas relying on the strictly convex and real-analytic cumulant function characterizing the exponential family. In this work, we report (dis)similarity formulas which bypass the explicit use of the cumulant function and highlight the role of quasi-arithmetic means and their multivariate mean operator extensions. In practice, these cumulant-free formulas are handy when implementing these (dis)similar…

Cumulants of Hawkes point processes

We derive explicit, closed-form expressions for the cumulant densities of a multivariate, self-exciting Hawkes point process, generalizing a result of Hawkes in his earlier work on the covariance density and Bartlett spectrum of such processes. To do this, we represent the Hawkes process in terms of a Poisson cluster process and show how the cumulant density formulas can be derived by enumerating all possible "family trees", representing complex interactions between point events. We also consider the problem of computing the integrated cumulants, characterizing the average measure of correlate…

多元正态性BHEP检验的四矩近似与快速$p$值计算

Four-Moment Approximations and Fast $p$-Values for BHEP Tests of Multivariate Normality

本文推导了BHEP多元正态性检验极限分布的第四累积量，并基于前四阶矩构造Johnson和Pearson近似，实现了无需特征值计算的快速精确$p$值计算。

通过模函数研究Hüsler-Reiss分布的极值条件独立性

Extremal conditional independence for Hüsler-Reiss distributions via modular functions

本文通过引入两个集合函数和模ularity关系，研究Hüsler-Reiss分布的极值条件独立性，并探讨其在高维极值统计中的应用。

Multivariate saddlepoint approximations in tail probability and conditional inference

We extend known saddlepoint tail probability approximations to multivariate cases, including multivariate conditional cases. Our approximation applies to both continuous and lattice variables, and requires the existence of a cumulant generating function. The method is applied to some examples, including a real data set from a case-control study of endometrial cancer. The method contains less terms and is easier to implement than existing methods, while showing an accuracy comparable to those methods.

Moment properties of multivariate infinitely divisible laws and criteria for self-decomposability

Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely divisible distribution and any positive real number $α$, an absolute moment of order $α$ relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding L$\acute{\rm e}$vy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, is given by Sato (1999, Theorem 25.3). We extend Ramachandran's theorem to the multivariate case, keeping in mind the immediate requir…