Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 1, 28--54

Specification Tests Indexed by Bandwidths

Holger Dette and Benjamin Hetzler, Ruhr-Universit\"at Bochum, Germany

SUMMARY. In this note, we consider several goodness-of-fit tests for model specification in nonparametric regression models which are based on kernel methods. In order to circumvent the problem of choosing a bandwidth for the corresponding test statistic, we propose to consider the statistics as stochastic processes indexed with bandwidths proportional to the asymptotically optimal bandwidth for the estimation of the regression function. We prove weak convergence of these processes to centered Gaussian processes and suggest to use functionals of these processes as test statistics for the problem of model specification. A bootstrap test is proposed to obtain a good approximation of the nominal level. The results are illustrated by means of a simulation study and the new test is compared with some of the currently available procedures.

AMS (2000) subject classification. Primary 62G10, 62G20.

Key words and phrases. Goodness-of-fit test, weak convergence, nonparametric regression, specification test, selection of smoothing parameters.

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