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Goodness‐of‐fit Tests Based on the Kernel Density Estimator
Author(s) -
CAO RICARDO,
LUGOSI GÁBOR
Publication year - 2005
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2005.00471.x
Subject(s) - mathematics , estimator , kernel density estimation , consistency (knowledge bases) , kolmogorov–smirnov test , goodness of fit , statistics , smoothing , kernel (algebra) , statistical hypothesis testing , combinatorics , discrete mathematics
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 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].