On Nonparametric Estimates of Probability Density and Its Derivatives

Nonparametric estimates for the probability distribution density and its derivatives up to third order inclusive are investigated. Estimates of both the probability density itself and of its derivatives are constructed by using weighted functions of various orders, the order being determined by conditions imposed on the moments. Using the example of two specific probability densities having derivatives of all orders, it is shown that the use of higher-order weighting functions leads to an impressive gain in accuracy for a sample of large volume, and, conversely, results in a loss in accuracy when the sample size is modest.