Wolfgang Wefelmeyer, University of Cologne, Germany.
TITLE:
Non-standard behavior of density estimators for sums of squares
Abstract
It has been shown recently that, under an appropriate moment condition,
densities of functions of independent and identically distributed random
variables can be estimated at the parametric rate by a local U-statistic.
For the sum of two squared random variables, the moment condition
typically fails. We show that then the local U-statistic
behaves differently at different points.
At points in the support of the squared random variable,
the rate of the local U-statistic slows down by a logarithmic factor
and is independent of the bandwidth, but the asymptotic variance depends
on the rate of the bandwidth, and otherwise only on the density
of the squared random variable at this point and at zero.
Of course, for bounded random variables, the sum of squares is
more spread out than a single square. At points outside the support
of the squared random variable, the local U-statistic behaves classically.
Now the rate is again parametric, the asymptotic variance
has a different form and does not depend on the bandwidth.
This is joint work with Anton Schick.
