On Admissible Nonparametric Estimates of the Probability Density and its Derivatives

Nonparametric estimates (estimates of the Rosenblatt–Parzen type) for an unknown probability density $f(x)$ are considered. Particular attention is paid to estimates using the admissible weighting functions $K(x)$, i.e., the weighting functions which possess the properties of having Fourier transforms that are non negative and not exceeding the value $\Psi(t)$. Some examples of admissible weighting functions $K(x)$ are presented. Estimates for the functions $f(x)$ of orders $\gamma=1$ and 2 are considered from a similar point of view.