Local linear smoothers using inverse Gaussian regression

Abstract

Local linear fitting has been widely used for estimating the univariate regression function, it has numerous fantastic properties like minimax efficiency and boundary correction. The asymmetric kernel functions match the support of the explanatory variables, and we discover the inverse Gaussian (IG) kernel function is identical to the normal kernel with variable bandwidth. Based on these, this paper proposes a new local linear estimation procedure with the IG kernel when the covariates are supported on \((0,\infty )\). Asymptotic theories of the proposed estimator are systematically studied, including the conditional mean squared error, the asymptotic normality and the uniform almost sure convergence. A simulation study and a data example indicate that the proposed estimator works efficiently.