This paper by Ruppert and Wand extends the asymptotic bias and variance analysis of multivariate local regression estimators to kernel weights. The authors derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory, which is particularly useful for analyzing the asymptotic conditional bias and variance at points near the boundary of the predictor variable support. They also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator and higher-order polynomial fits and derivative estimation in the univariate case. The main findings include the derivation of asymptotic bias and variance expressions for local polynomial fitting, which are shown to be superior to those of the Nadaraya–Watson or Gasser–Müller estimators, especially near the boundary of the support. The paper provides a comprehensive analysis of the asymptotic behavior of these estimators, highlighting their advantages and limitations.This paper by Ruppert and Wand extends the asymptotic bias and variance analysis of multivariate local regression estimators to kernel weights. The authors derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory, which is particularly useful for analyzing the asymptotic conditional bias and variance at points near the boundary of the predictor variable support. They also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator and higher-order polynomial fits and derivative estimation in the univariate case. The main findings include the derivation of asymptotic bias and variance expressions for local polynomial fitting, which are shown to be superior to those of the Nadaraya–Watson or Gasser–Müller estimators, especially near the boundary of the support. The paper provides a comprehensive analysis of the asymptotic behavior of these estimators, highlighting their advantages and limitations.