Non-Standard Behavior of Density Estimators for Functions of Independent Observations

Densities of functions of two or more independent random variables can be estimated by local U-statistics. Frees (1994) gives conditions under which they converge pointwise at the parametric root-n rate. Uniform convergence at this rate was established by Schick and Wefelmeyer (2004) for sums of random variables. Giné and Mason (2007) give conditions under which this rate also holds in Lp-norms. We present several natural applications in which the parametric rate fails to hold in Lp or even pointwise. 1. The density estimator of a sum of squares of independent observations typically slows down by a logarithmic factor. For exponents greater than two, the estimator behaves like a classical density estimator. 2. The density estimator of a product of two independent observations typically has the root-n rate pointwise, but not in Lp-norms. An application is given to semi-Markov processes and estimation of an inter-arrival density that depends multiplicatively on the jump size. 3. The stationary density of a nonlinear or nonparametric autoregressive time series driven by independent innovations can be estimated by a local U-statistic (now based on dependent observations and involving additional parameters), but the root-n rate can fail if the derivative of the autoregression function vanishes at some point.