Limit distribution of integral functionals of nuclear estimates of probability densities (Q750042)

Let \(X_ 1,...,X_ n\) be an i.i.d. sample from a population with an unknown density f(x). Consider a kernel estimate of f(x): \[ f_ n(x)=(na_ n)^{-1}\sum^{n}_{i=1}W((x-X_ i)/a_ n). \] Denote for an integer \(p\geq 1:\) \[ B_ n^{2p}=\int (f_ n(x)-E f_ n(x))^{2p} h(x)dx, \] where h(x) is a weight function. Under some assumptions on f, w, h and \(a_ n\), the author proves that \(B_ n^{2p}\) has an asymptotic normal distribution. This result is based upon the methods and results of \textit{V. S. Korolyuk} and \textit{Yu. V. Borovskikh} [Asymptotic analysis of distributions in statistics. (Russian) (1984; Zbl 0565.62003)].