Uniform strong consistency of interpolated and smoothing S-statistics (Q1263190)

One of the possible approaches to constructing estimates of the distribution density of a random variable with the help of interpolated splines was proposed by \textit{L. I. Boneva}, \textit{D. Kendall} and \textit{I. Stefanov} [J. R. Stat. Soc., Ser. B 33, 1-70 (1971; Zbl 0231.62004)]. In \textit{G. Wahba}, Ann. Stat. 3, 30-48 (1975; Zbl 0305.62022), based on this approach, uniform (in some class of distributions) mean-square convergence was shown of an interpolated spline-approximation of an empirical density and an estimate of the speed of convergence was obtained. However, the established speed of convergence does not ensure strong consistency. Here an approach somewhat different from that offered by Boneva et al. to constructing density estimates of a random variable is proposed. S- statistics are taken as such estimates, constructed both for interpolated and smoothing splines and being, in turn, densities of some random variables, and their uniform strong consistency on the interval is shown.