Estimating regression functions and their derivatives by the kernel method

Abstract

A kernel estimate is introduced for obtaining a nonparametric estimate of a regression function, as well as of its derivatives. In many fields of engineering and biomedicine, the estimation of velocity and acceleration is of great importance in addition to obtaining a smoothed curve from the measurements themselves. Some asymptotic properties, as, for example, weak and strong consistency and asymptotic normality are derived, as well as asymptotic expressions for bias and variance. A solution for treating the extremities of the data is investigated in order to obtain the asymptotic integrated mean squared error, and also in order to facilitate data analysis over the range of abscissae investigated. Some finite sample results, and two examples of application in biomedicine, render some insight into the potential usefulness of this approach.