Bandwidth selection in local polynomial regression using eigenvalues

Local polynomial regression is commonly used for estimating regression functions. In practice, however, with rough functions or sparse data, a poor choice of bandwidth can lead to unstable estimates of the function or its derivatives. We derive a new expression for the leading term of the bias by using the eigenvalues of the weighted design matrix where the bias depends on the arrangement of the "X"-values in the bandwidth window. We then use this result to determine a local data-driven bandwidth selection method and to provide a diagnostic for poor bandwidths that are chosen by using other methods. We show that our data-driven bandwidth is asymptotically equivalent to the optimal local bandwidth and that it performs well for relatively small samples when compared with other methods. In addition, we provide simulation results for first-derivative estimation. We illustrate its performance with data from Mars Global Surveyor. Copyright 2006 Royal Statistical Society.