Local linear estimation of covariance matrices via Cholesky decomposition

A fundamental problem in multivariate statistics is the estimation of covariance matrices. We consider in this paper a flexible class of nonparametric covariance models for which the entries in the covariance matrix depend on covariates. Although it is well known that local linear estimation is much preferred over local constant estimation (Fan & Gijbels, 1996), developing the former for estimating covariance matrices is challenging due to the positive definiteness constraint of such matrices. Motivated by the modified Cholesky decomposition, we propose for the first time a local linear estimator of the nonparametric covariance matrix. The proposed estimator is positive definite, is adaptive, possesses good theoretical properties and performs well in numerical studies. An application to the Boston housing data is also provided to illustrate attractive finite-sample performance of the proposed method.