Contributions To The Statistical Inference For The Semiparametric Elliptical Copula Model
This thesis addresses aspects of the statistical inference problem for the semiparametric elliptical copula model. A copula (function) for a continuous multivariate distribution is the joint distribution function of the transformed marginal distributions, where the transformation is the probability integral transform. As such, copula is a tool to couple or decouple the multivariate dependence structure from the behaviors of the individual margins. The semiparametric elliptical copula model is the family of distributions whose dependence structures are specified by parametric elliptical copulas but whose marginal distributions are left unspecified. The elliptical copula is in turn uniquely characterized by a characteristic generator and a copula correlation matrix [SIGMA]. In the first part of this thesis, we address the estimation of [SIGMA]. A natural estimate for [SIGMA] is the plug-in estimator [SIGMA] with Kendall's tau statistic. We first obtain a sharp bound on the operator norm of [SIGMA] [-] [SIGMA]. Then, we study a factor model of [SIGMA], for which we propose a refined estimator [SIGMA] by fitting a lowrank matrix plus a diagonal matrix to [SIGMA] using least squares with a nuclear norm penalty on the low-rank matrix. The bound on the operator norm of [SIGMA] [-] [SIGMA] serves to scale the penalty term, and we obtain finite sample oracle inequalities for [SIGMA]. We provide data-driven versions of all our estimation procedures. In the second part of this thesis, we specialize to a subset of the semiparametric elliptical copula model and study the classification of two distributions that have the same Gaussian copula but that are otherwise arbitrary in high dimensions. Under this semiparametric Gaussian copula setting, we derive an accurate semiparametric estimator of the log density ratio, which leads to our empirical decision rule and a bound on its associated excess risk. Our estimation procedure takes advantage of the potential sparsity as well as the low noise condition in the problem, which allows us to achieve faster convergence rate of the excess risk than is possible in the existing literature on semiparametric Gaussian copula classification. We demonstrate the efficiency of our semiparametric empirical decision rule by showing that the bound on the excess risk nearly achieves a convergence rate of n[-]1/2 in the simple setting of Gaussian distribution classification.
semiparametric; elliptical copula
Bunea,Florentina; Wells,Martin Timothy
Ph. D., Statistics
Doctor of Philosophy
dissertation or thesis