New Models For Conditional Covariance Matrix

Open Access
Zhang, Ying
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
January 30, 2013
Committee Members:
  • Runze Li, Dissertation Advisor
  • Runze Li, Committee Chair
  • Bing Li, Committee Member
  • Zhibiao Zhao, Committee Member
  • Qiang Du, Committee Member
  • covariance estimation
  • smoothing spline
  • Cholesky decomposition
  • local linear regression
  • semi-parametric regression
This thesis presents new models for conditional covariance matrix. The proposed non- parametric covariance regression model parameterizes the conditional covariance matrix of a multivariate response vector as a quadratic function of regression splines. The re- sulting conditional covariance matrix is positive definite for all explanatory variables and represents the conditional covariance as the summation of a “baseline” covariance matrix and a positive definite matrix depending on the explanatory variables. The pro- posed approach provides an adaptable representation of heteroscedasticity across the levels of explanatory variable. In addition, the model has a random-effect representa- tion, allowing for the maximum likelihood parameter estimation via the EM-algorithm. The asymptotic normality for the estimators is established and some numerical examples are used to illustrate the proposed procedure. To cope with the high-dimensionality of the covariates, estimating the conditional covariance matrix through a modified Cholesky decomposition is proposed. The mod- ified Cholesky decomposition procedure associates each local covariance matrix with a unique unit lower triangular and a unique diagonal matrix. The entries of the lower triangular matrix and the diagonal matrix have statistical interpretation as regression co- efficients and prediction variances when regressing each term on its predecessors. It ensures that the estimated conditional covariance matrix is positive definite. To circum- vent the curse of dimensionality, a class of partially linear models are used to estimate those regression coefficients and local linear estimators are developed to estimate the nonparametric variance functions. The asymptotic properties of the proposed procedure are studied. We show that the proposed procedure for estimating the conditional covari- ance matrix based on residuals has the same asymptotic bias and variance as that based on true errors. Comprehensive simulation studies and a real data example are presented to illustrated the proposed methods.