Local Likelihood Estimation

Abstract A scatterplot smoother is applied to data of the form {(x 1, y 1), (x 2, y 2, …, (xn, yn )} and uses local fitting to estimate the dependence of Y on X. A simple example is the running lines smoother, which fits a least squares line to the y values falling in a window around each x value. The value of the estimated function at x is given by the value of the least squares line at x. A smoother generalizes the least squares line, which assumes that the dependence of Y on X is linear. In this article, we extend the idea of local fitting to likelihood-based regression models. One such application is to the class of generalized linear models (Nelder and Wedderburn 1972). We enlarge this class by replacing the covariate form β0 + xβ1 with an unspecified smooth function s(x). This function is estimated from the data by a technique we call local likelihood estimation. The method consists of maximum likelihood estimation for β0 and β1, applied in a window around each x value. Multiple covariates are incor...