Convergence rates of density estimators for sums of powers of observations

Abstract

Densities of functions of independent and identically distributed random observations can be estimated by a local U-statistic. It has been shown recently that, 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 two. We have shown elsewhere that for exponent equal to two the rate of convergence slows down by a logarithmic factor on the support of the squared observation and is still parametric outside this support. For exponent greater than two, and on the support of the exponentiated observation, the estimator behaves like a classical density estimator: The bias is not negligible and the rate depends on the bandwidth. Outside the support, the rate is again parametric.