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 incorporated through an iterative algorithm. We also apply the local likelihood technique to the proportional hazards model of Cox (1972), a model for censored data. The proportional hazards assumption λ (t | x) = λ0 (t) exp (xβ) is replaced by λ (t | x) = λ0 (t) exp (s (x) ), and the function s (x) is estimated from the data by the local likelihood method. In some real data examples, the local likelihood technique proves to be effective in uncovering nonlinear dependencies. It is useful as a descriptive tool or to suggest transformations of the covariates. We also discuss some methods for inference.