On the conditional distribution of goodness-of-fit tests

This manuscript advocates the use of the conditional distribution of the goodness-of-fit test, given the value of the minimal sufficient statistic for the parameters, in the problem of testing the fit of a distribution known only in its form. In such a setting, since the parameters themselves are not of interest, they are considered nuisance and so conditioning seems to be appropriate. Some comments are made regarding this procedure and emphasis is placed on the fact that with this approach, there is no need for sets of tables but rather for just an algorithm, based on a special simulation which produces the '' exact '' conditional p-value. So it may be claimed to be an exact level alpha, finite-n procedure, in the continuous case. It may be used in the discrete case also but the level would be approximate. As an example, the inverse Gaussian is discussed, comparing the results of the proposed procedure with recent work, by means of some simulations, showing that for the alternatives studied, there is an increase of power.