On the integral mean squared error of some non-parametric estimates of the probability density

It is shown that in estimating the density $p(x)$ by means of the statistics (1) the sequence $\tau_n=\tau_n^0$ is optimal in the sense of the minimum integral mean squared error $U_n^2(\tau_n)$. An estimate $\widehat\tau_n=\widehat\tau_n(X_1, X_2,\dots,X_n)$ for $\tau_n^0$ is constructed and a theorem is proved that gives conditions under which $U_n^2(\widehat\tau_n)\sim U_n^2(\tau_n^0)$.