Comparisons of B-spline procedures with kernel procedures in estimating regression functions and their derivatives

Abstract

There are several methods to estimate regression functions and their derivatives. Among them, B-spline procedures and kernel procedures are known to be useful. However, at present, it is not determined which procedure is better than the others. In this paper, we investigate the performance of the procedures by computer simulations. Two B-spline procedures are considered. The first one is to estimate derivatives using a different roughness penalty for each degree of the derivative d. In this procedure, the smoothing parameters and the coefficients of the B-spline functions are different for each d. The second procedure is to estimate the dth derivative just by differentiating the estimated regression function d-times. In this case, the regression function and its derivatives have a common coefficient vector of B-spline functions. Two kernel procedures are also considered. The first kernel procedure used in our simulations is constructed with the Gasser-Muller estimator and a global plug-in bandwidth selector. The second one is a local polynomial fitting with a refined bandwidth selector. As a result of our simulations, we find that B-spline procedures can give better estimates than the kernel ones in estimating regression functions. For derivatives, we also find that in B-spline methods, it is necessary to choose a different smoothing parameter (or coefficient vector) for each degree of derivative; between the two kernel methods, the Gasser-Muller procedure gives better results than the local polynomial fitting in most cases. Furthermore, the first B-spline method can still work better than the Gasser-Muller procedure in the central area of the domain of the functions. But in the boundary areas, the Gasser-Muller procedure gives more stable derivative estimates than all the other methods.