Stein's method on Wiener chaos

We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-It\^o integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Ess\'een bounds in the Breuer-Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.Comment: 39 pages; Two sections added; To appear in PTR

Statistical learning techniques applied to epidemiology: a simulated case-control comparison study with logistic regression

<p>Abstract</p> <p>Background</p> <p>When investigating covariate interactions and group associations with standard regression analyses, the relationship between the response variable and exposure may be difficult to characterize. When the relationship is nonlinear, linear modeling techniques do not capture the nonlinear information content. Statistical learning (SL) techniques with kernels are capable of addressing nonlinear problems without making parametric assumptions. However, these techniques do not produce findings relevant for epidemiologic interpretations. A simulated case-control study was used to contrast the information embedding characteristics and separation boundaries produced by a specific SL technique with logistic regression (LR) modeling representing a parametric approach. The SL technique was comprised of a kernel mapping in combination with a perceptron neural network. Because the LR model has an important epidemiologic interpretation, the SL method was modified to produce the analogous interpretation and generate odds ratios for comparison.</p> <p>Results</p> <p>The SL approach is capable of generating odds ratios for main effects and risk factor interactions that better capture nonlinear relationships between exposure variables and outcome in comparison with LR.</p> <p>Conclusions</p> <p>The integration of SL methods in epidemiology may improve both the understanding and interpretation of complex exposure/disease relationships.</p

Characterizations of mixtures of discrete distributions by a regression point

Given that the conditional distribution ps(y|x) of Y, given X = x is an x-fold convolution of a nonnegative integer-valued r.v. ξ for every s= P[ξ = 0] &amp;gt; 0, the distribution of X, hence also of Y, is characterized by the regression point m(0) = E[X|Y = 0]. An infinite variety of generalized distributions (of Y) can be characterized by arbitrarily varying the distribution of X. © 1983

Two LDF characterizations of the normal as a spherical distribution

Two optimal characteristic properties of the normal distribution are shown: (a) Of all the SNM (spherical scale normal mixtures) the normal with the same Mahalanobis distances between Πi:SNM(μi) and Πj:SNM(μj), i ≠ j, maximizes the probabilities of correct classification determined by a certain subclass of the LDF classification rules; (b) The class of LDF (linear discriminant function) rules is the admissible class for the discrimination problem with spherical population alternatives iff the spherical distribution is normal. © 1992

The F -test of homoscedasticity for correlated normal variables

Using the F-representation of t, the Pitman-Morgan t-test for homoscedasticity under a bivariate normal setup is shown to be equivalent to an F-test on n-2 and n-2 degrees of freedom. This yields an F-test of independence under normality. © 2001 Elsevier Science B.V