DATA-DRIVEN BANDWIDTH SELECTION IN LOCAL POLYNOMIAL FITTING - VARIABLE BANDWIDTH AND SPATIAL ADAPTATION

When estimating a mean regression function and its derivatives, locall y weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimatin g curves with a complicated structure, a variable bandwidth is desirab le. Furthermore, the bandwidth should be indicated by the data themsel ves. Recent developments in nonparametric smoothing techniques inspire d us to propose such a data-driven bandwidth selection procedure, whic h can be used to select both constant and variable bandwidths. The ide a is based on a residual squares criterion along with a good approxima tion of the bias and variance of the estimator. The procedure can be a pplied to select bandwidths not only for estimating the regression cur ve but also for estimating its derivatives. The resulting estimation p rocedure has the necessary flexibility for capturing complicated shape s of curves. This is illustrated via a large variety of testing exampl es, including examples with a large spatial variability. The results a re also compared with wavelet thresholding techniques, and it seems th at our results are at least comparable, i.e. local polynomial regressi on using our data-driven variable bandwidth has spatial adaptation pro perties that are similar to wavelets.