Multivariate Local Polynomial Kernel Estimators: Leading Bias and Asymptotic Distribution

Masry (1996b) provides estimation bias and variance expression for a general local polynomial kernel estimator in a general multivariate regression framework. Under smoother conditions on the unknown regression function and by including more refined approximation terms than that in Masry (1996b), we extend the result of Masry (1996b) to obtain explicit leading bias terms for the whole vector of the local polynomial estimator. Specifically, we derive the leading bias and leading variance terms of nonparametric local polynomial kernel estimator in a general nonparametric multivariate regression model framework. The results can be used to obtain optimal smoothing parameters in local polynomial estimation of the unknown conditional mean function and its derivative functions.