On efficient estimation of densities for sums of squared observations

Densities of functions of independent and identically distributed random observations can be estimated by using a local U-statistic. Under an appropriate integrability condition, this estimator behaves asymptotically like an empirical estimator. In particular, it converges at the parametric rate. The integrability condition is rather restrictive. It fails for the sum of powers of two observations when the exponent is at least 2. We have shown elsewhere that for the exponent equal to 2 the rate of convergence slows down by a logarithmic factor in the support of the squared observation. Here we show that the estimator is efficient in the sense of Hájek and Le Cam. In particular, the convergence rate is optimal.