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

Abstract

Abstract. 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 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 L2 density‐based approach due to Fan [Econ. Theory10 (1994) 316].