arXiv subjects

Donald Richards

Publications and source records attributed to Donald Richards.

40 records · Page 3Linked to original sources

arXiv2015

Gaussian Random Particles with Flexible Hausdorff Dimension

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises--Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.

math.PR↗

arXiv2014

A Generalization of an Integral Arising in the Theory of Distance Correlation

We generalize an integral which arises in several areas in probability and statistics and which is at the core of the field of distance correlation, a concept developed by Székely, Rizzo and Bakirov (2007) to measure dependence between random variables. Let $m$ be a positive integer and let ${\cos_m}(u)$, $u \in \mathbb{R}$, be the truncated Maclaurin expansion of ${\cos}(u)$, where the expansion is truncated at the $m$th summand. For $t, x \in \mathbb{R}^d$, let $\langle t,x\rangle$ and $\|x\|$ denote the standard Euclidean inner product and norm, respectively. We establish the integral formula: For $α\in \mathbb{C}$ and $x \in \mathbb{R}^d$, $\int_{\mathbb{R}^d} [\cos_m(\langle t,x\rangle) - \cos(\langle t,x\rangle)] \,{\rm d}t/{\|t\|^{d+α}} = C(d,α) \, \|x\|^α$, with absolute convergence if and only if $2(m-1) < \Re(α) < 2m$. Moreover, the constant $C(d,α)$ does not depend on $m$.

math.ST↗

arXiv2014

The affinely invariant distance correlation

Székely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769-2794) and Székely and Rizzo (Ann. Appl. Statist. 3 (2009) 1236-1265), in two seminal papers, introduced the powerful concept of distance correlation as a measure of dependence between sets of random variables. We study in this paper an affinely invariant version of the distance correlation and an empirical version of that distance correlation, and we establish the consistency of the empirical quantity. In the case of subvectors of a multivariate normally distributed random vector, we provide exact expressions for the affinely invariant distance correlation in both finite-dimensional and asymptotic settings, and in the finite-dimensional case we find that the affinely invariant distance correlation is a function of the canonical correlation coefficients. To illustrate our results, we consider time series of wind vectors at the Stateline wind energy center in Oregon and Washington, and we derive the empirical auto and cross distance correlation functions between wind vectors at distinct meteorological stations.

math.ST↗

arXiv2014

EM algorithms for estimating the Bernstein copula

A method that uses order statistics to construct multivariate distributions with fixed marginals and which utilizes a representation of the Bernstein copula in terms of a finite mixture distribution is proposed. Expectation-maximization (EM) algorithms to estimate the Bernstein copula are proposed, and a local convergence property is proved. Moreover, asymptotic properties of the proposed semiparametric estimators are provided. Illustrative examples are presented using three real data sets and a 3-dimensional simulated data set. These studies show that the Bernstein copula is able to represent various distributions flexibly and that the proposed EM algorithms work well for such data.

stat.CO↗