Local linear estimation of covariance matrices via cholesky decomposition

An important problem in multivariate statistics is the estimation of covariance matrices. We consider a class of nonparametric covariance models in which the entries in the covariance matrix depend on covariates. Previously, the locally constant approach was used for estimating this matrix due to its simplicity. However, to ensure the positive definiteness of the resulting estimator, a single bandwidth parameter was used for estimating all the elements in this matrix. We propose to use the locally linear method, a technique known to outperform local constant estimation, for estimating the elements after the modified Cholesky decomposition. The proposed estimator is guaranteed to be positive definite, allows different degrees of smoothing for different elements, possesses good theoretical properties, and performs well in numerical studies. An application to the Boston housing data is provided to illustrate the finite-sample performance of the proposed method.