Goodness of fit for the constancy of a classical statistical model over time

The classical statistical model relates to n independent random variables having a common distribution. In this paper we consider the situation where the common distribution involves an unknown parameter, and where at time 0<t <1 only the first [nt] random variables are observed. The innovation approach is used to derive goodness of fit processes which especially detect alternatives under which the unknown parameter does not remain constant, but varies over time. The behaviour of these processes is investigated under the null hypothesis as well as under alternative hypotheses. Limiting Pitman efficacies of supremum type tests based on these processes are evaluated. Fixed change point alternative hypotheses and smooth alternative hypotheses receive additional treatment. The methods are exemplified using covariance structure models, especially Gaussian graphical models.