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 L1-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 nonasymptotic 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 L2 density-based approach due to Fan [Econ. Theory 10 (1994) 316].