Statistical Estimation of the Derivatives of Density Functions

THE need to estimate the slope (or higher derivatives) of a probability density function from a sample of independent observations has suggested an extension of a technique previously suggested1 for estimating density functions. The proposed extension is straightforward, and is summarized here for the first derivative. Let the sample cumulative distribution function be Fs(u), and let the true density function be f(u), with successive derivatives f1(u),f2(u), …. Let the estimate of f1(u) at x be