This paper discusses the asymptotic properties of multivariate locally weighted least squares regression estimators. It extends previous results on the asymptotic bias and variance of kernel estimators to the multivariate case. The authors analyze the local linear kernel-weighted least squares regression estimator, which is shown to have superior asymptotic properties compared to the Nadaraya–Watson and Gasser–Müller estimators. The paper also investigates the asymptotic properties of the multivariate local quadratic least squares regression estimator and higher-order polynomial fits, as well as derivative estimation. The paper begins by introducing nonparametric regression and its importance in modeling real data. It then presents the multivariate nonparametric regression problem, which involves estimating the conditional mean $ m(x) = E(Y \mid X = x) $ without assuming a parametric form for $ m $. The paper discusses the use of kernel estimators, which are simple to understand and implement, and their asymptotic properties. It also highlights the advantages of local linear least squares estimators, which have been shown to have better asymptotic properties than other kernel estimators. The paper derives the asymptotic bias and variance of the multivariate local regression estimator. It shows that the leading terms of the bias and variance depend on the bandwidth matrix $ H $, which controls the amount of smoothing in each direction. The paper also discusses the asymptotic properties of the estimator at boundary points, where the density of the predictors is lower, and shows that the bias and variance are of the same order of magnitude as in the interior. The paper also extends the results to higher-degree polynomial fits and derivative estimation. It shows that the asymptotic bias and variance of the local polynomial estimator depend on the order of the polynomial and the bandwidth matrix. The paper concludes by discussing the importance of understanding the asymptotic properties of nonparametric regression estimators, particularly in the context of multivariate data and boundary regions.This paper discusses the asymptotic properties of multivariate locally weighted least squares regression estimators. It extends previous results on the asymptotic bias and variance of kernel estimators to the multivariate case. The authors analyze the local linear kernel-weighted least squares regression estimator, which is shown to have superior asymptotic properties compared to the Nadaraya–Watson and Gasser–Müller estimators. The paper also investigates the asymptotic properties of the multivariate local quadratic least squares regression estimator and higher-order polynomial fits, as well as derivative estimation. The paper begins by introducing nonparametric regression and its importance in modeling real data. It then presents the multivariate nonparametric regression problem, which involves estimating the conditional mean $ m(x) = E(Y \mid X = x) $ without assuming a parametric form for $ m $. The paper discusses the use of kernel estimators, which are simple to understand and implement, and their asymptotic properties. It also highlights the advantages of local linear least squares estimators, which have been shown to have better asymptotic properties than other kernel estimators. The paper derives the asymptotic bias and variance of the multivariate local regression estimator. It shows that the leading terms of the bias and variance depend on the bandwidth matrix $ H $, which controls the amount of smoothing in each direction. The paper also discusses the asymptotic properties of the estimator at boundary points, where the density of the predictors is lower, and shows that the bias and variance are of the same order of magnitude as in the interior. The paper also extends the results to higher-degree polynomial fits and derivative estimation. It shows that the asymptotic bias and variance of the local polynomial estimator depend on the order of the polynomial and the bandwidth matrix. The paper concludes by discussing the importance of understanding the asymptotic properties of nonparametric regression estimators, particularly in the context of multivariate data and boundary regions.