On a class ol nonparametric estimates of nonlinear functionals of a density

A new class of nonparametric estimated of functionals of the type $J=\int F(g(x))\,dx$ (where $F(\,\cdot\,)$ is a certain function and $g(x)$ is the unknown density function) is proposed. The estimates are based on combination of Rosenblatt–Parzen's estimate of the density and Hoeffing's $U$-statistic. They are called quasi-$U$-statistics. The performance of quasi-$U$-statistics is demonstrated by an example of estimation of the squared density integral.