Literature Review for Local Polynomial Regression

This paper discusses key results from the literature in the field of local polynomial regression. Local polynomial regression (LPR) is a nonparametric technique for smoothing scatter plots and modeling functions. For each point, x0, a low-order polynomial WLS regression is fit using only points in some “neighborhood” of x0. The result is a smooth function over the support of the data. LPR has good performance on the boundary and is superior to all other linear smoothers in a minimax sense. The quality of the estimated function is dependent on the choice of weighting function, K, the size the neighborhood, h, and the order of polynomial fit, p. We discuss each of these choices, paying particular attention to bandwidth selection. When choosing h, “plug-in” methods tend to outperform cross-validation methods, but computational considerations make the latter a desirable choice. Variable bandwidths are more flexible than global ones, but both can have good asymptotic and finite-sample properties. Odd-order polynomial fits are superior to even fits asymptotically, and an adaptive order method that is robust to bandwidth is discussed. While the Epanechnikov kernel is the best in asymptotic minimax sense, a variety of kernels are used in practice. Extensions to various types of data and other applications of LPR are also discussed.