Date: 2015-09-25

Time: 15:30-16:30

Location: BURN 1205

Abstract:

This talk addresses aspects of the statistical inference problem for the semiparametric elliptical copula model. 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. An elliptical copula is uniquely characterized by a characteristic generator and a copula correlation matrix Sigma. In the first part of this talk, I will consider the estimation of Sigma. A natural estimate for Sigma is the plug-in estimator Sigmahat with Kendall’s tau statistic. I will first exhibit a sharp bound on the operator norm of Sigmahat - Sigma. I will then consider a factor model of Sigma, for which I will propose a refined estimator Sigmatilde by fitting a low-rank matrix plus a diagonal matrix to Sigmahat using least squares with a nuclear norm penalty on the low-rank matrix. The bound on the operator norm of Sigmahat - Sigma serves to scale the penalty term, and we obtained finite-sample oracle inequalities for Sigmatilde that I will present. In the second part of this talk, we will look at 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, I will give an accurate semiparametric estimator of the log-density ratio, which leads to an 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. I will demonstrate the efficiency of our semiparametric empirical decision rule by showing that the bound on the excess risk nearly achieves a convergence rate of 1 over square-root-n in the simple setting of Gaussian distribution classification.

Speaker

Yue Zhao is a Postdoctoral Fellow in the Department of Mathematics and Statistics at McGill University, Montréal.