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.

In this paper a general method of constructing control charts for preliminary analysis of individual observations is presented, which is based on recursive score residuals. A simulation study shows that certain implementations of these charts are highly effective in detecting assignable causes.

Suppose that a sequence of data points follows a distribution of a certain parametric form, but that one or more of the underlying parameters may change over time. This paper addresses various natural questions in such a framework. We construct canonical monitoring processes which under the hypothesis of no change converge in distribution to independent Brownian bridges, and use these to construct natural goodness-of-fit statistics. Weighted versions of these are also studied, and optimal weight functions are derived to give maximum local power against alternatives of interest. We also discuss how our results can be used to pinpoint where and what type of changes have occurred, in the event that initial screening tests indicate that such exist. Our unified large-sample methodology is quite general and applies to all regular parametric models, including regression, Markov chain and time series situations.