Local Likelihood Estimation

Abstract

A scatterplot smoother is applied to data of the form {(x ₁, y ₁), (x ₂, y ₂, …, (x_n, y_n )} 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.

Key Words:

• Smoothing
• Generalized linear models
• Nonparametric regression