arXiv subjects
Publications and source records attributed to Gábor J. Székely.
In this paper three unrelated problems will be discussed. What connects them is the rich methodology of classical probability theory. In the first two problems we have a complete answer to the problems raised; in the third case, what we call the Hungarian roulette problem, we only have a conjecture with heuristic justification.
Since Pearson's correlation was introduced at the end of the 19th century many dependence measures have appeared in the literature. Recently we have suggested four simple axioms for dependence measures of random variables that take values in Hilbert spaces. We showed that distance correlation satisfies all these axioms. We still need a new measure of dependence because existing measures either do not work in general metric spaces (that are not Hilbert spaces) or they do not satisfy our four simple axioms. The earth mover's correlation introduced in this paper applies in general metric spaces and satisfies our four axioms (two of them in a weaker form).
This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where the uncertainties in the major stochastic steps are quantified. Such stochastic steps involve approximating the NLS objective function using Monte-Carlo methods, and this is equivalent to the estimation of the trace of corresponding symmetric positive semi-definite (SPSD) matrices. For the latter, we prove tight necessary and sufficient conditions on the sample size (which translates to cost) to satisfy the prescribed probabilistic accuracy. We show that these conditions are practically computable and yield small sample sizes. They are then incorporated in our stochastic algorithm to quantify the uncertainty in each randomized step. The bounds we use are applications of more general results regarding extremal tail probabilities of linear combinations of gamma distributed random variables. We derive and prove new results concerning the maximal and minimal tail probabilities of such linear combinations, which can be considered independently of the rest of this paper.
In classical analysis of variance, dispersion is measured by considering squared distances of sample elements from the sample mean. We consider a measure of dispersion for univariate or multivariate response based on all pairwise distances between-sample elements, and derive an analogous distance components (DISCO) decomposition for powers of distance in $(0,2]$. The ANOVA F statistic is obtained when the index (exponent) is 2. For each index in $(0,2)$, this decomposition determines a nonparametric test for the multi-sample hypothesis of equal distributions that is statistically consistent against general alternatives.
Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the definition with identity. The corresponding statistic has an elegantly simple computing formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated.
Rejoinder to "Brownian distance covariance" by Gábor J. Székely and Maria L. Rizzo [arXiv:1010.0297]
Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented.
Generalized t-tests are constructed under weaker than normal conditions. In the first part of this paper we assume only the symmetry (around zero) of the error distribution (i). In the second part we assume that the error distribution is a Gaussian scale mixture (ii). The optimal (smallest) critical values can be computed from generalizations of Student's cumulative distribution function (cdf), $t_n(x)$. The cdf's of the generalized $t$-test statistics are denoted by (i) $t_n^S(x)$ and (ii) $t_n^G(x)$, resp. As the sample size $n\to \infty$ we get the counterparts of the standard normal cdf $Φ(x)$: (i) $Φ^S(x):=\operatorname {lim}_{n\to \infty}t_n^S(x)$, and (ii) $Φ^G(x):=\operatorname {lim}_{n\to \infty}t_n^G(x)$. Explicit formulae are given for the underlying new cdf's. For example $Φ^G(x)=Φ(x)$ iff $|x|\ge \sqrt{3}$. Thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of $Φ^G$ is $4\sqrt{3}/5=1.385... >Φ^{-1}(0.9)=1.282...$.