On nonparametric estimation of a functional of a probability density

The nonparametric estimate derived from the Hermite orthogonal system of the functional
I=\int f^{2}(x) dx
where
f
is an unknown probability density, is studied. Sufficient conditions for the weak and strong consistency of the estimate are presented, and the rate of convergence is given. In particular, under mild assumptions on
f
, the rate of mean-square error convergence is
O(n^{-1})
, whereas for almost complete convergence it is
O((n^{-1} \log n)^{1/2})
. Moreover, several possible applications in the area of nonparametric inference of the estimate are indicated.