Goodness-of-Fit Tests Based on the Kernel Density Estimate

Given an i.i.d. sample drawn from a density f on the real line, one has to test whether f is in a given class of densities. We investigate testing procedures constructed on the basis of minimizing the L 1 -distance between the kernel density estimate and any density in the hypothesized class. General nonasymptotic bounds are derived for the power of the test. It is shown that the concentratedness of the data-dependent smoothing factor plays a key role in the performance of the test, as well as the "size" of the hypothesized class of densities