Integrated Square Error Asymptotics for Supersmooth Deconvolution

We derive the asymptotic distribution of the integrated square error of a deconvolution kernel density estimator in supersmooth deconvolution problems. Surprisingly, in contrast to direct density estimation as well as ordinary smooth deconvolution density estimation, the asymptotic distribution is no longer a normal distribution but is given by a normalized chi-squared distribution with 2 d.f. A simulation study shows that the speed of convergence to the asymptotic law is reasonably fast. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..

Goodness-of-fit Tests Based on the Kernel Density Estimator

Given an i.i.d. sample drawn from a density "f" on the real line, the problem of testing whether "f" is in a given class of densities is considered. Testing procedures constructed on the basis of minimizing the "L" 1-distance between a kernel density estimate and any density in the hypothesized class are investigated. General non-asymptotic bounds are derived for the power of the test. It is shown that the concentration of the data-dependent smoothing factor and the 'size' of the hypothesized class of densities play a key role in the performance of the test. Consistency and non-asymptotic performance bounds are established in several special cases, including testing simple hypotheses, translation/scale classes and symmetry. Simulations are also carried out to compare the behaviour of the method with the Kolmogorov-Smirnov test and an "L" 2 density-based approach due to Fan ["Econ. Theory" 10 (1994) 316]. Copyright 2005 Board of the Foundation of the Scandinavian Journal of Statistics..