| MSC: | 62E17, 62E20 | |
Abstract:
Goodness of fit tests based on the empirical distribution function fail to be asymptotically distribution free if parameters are estimated. Asymptotically, the empirical process with estimated parameters is a centered Gaussian process with a covariance that differs from the covariance function of the Brownian bridge. If the maximum likelihood estimator is used then the new covariance function is smaller, in the Loewner semiorder, than the covariance function of the Brownian bridge. Therefore one may transform the empirical process with estimated parameters back to a Brownian bridge by adding an independent process that is suitably constructed. Classical goodness of fit statistics have aftzer this transformation an asymptotic distribution as in the case of known parameters. The power under local alternatives of this new goodness of fit tests is studied and computer simulationscompare the new test with their bootstrap versions.