Goodness of fit problem and scanning innovation martingales.

*(English)*Zbl 0801.62043In Theory Probab. Appl. 26, 240-257 (1982); translation from Teor. Veroyatn. Primen. 26, 246-265 (1981; Zbl 0454.60049), the author introduced the innovation process for the parametric empirical process in the case when both the i.i.d. random variables and the parameter are one- dimensional. This article extends these ideas to the case where the i.i.d. random variables are \(m\)-dimensional vectors and the parameter involved is \(d\)-dimensional.

In the first part, the author gives a short history of goodness of fit theory recalling the classical results of Kolmogorov and Doob. He also induces the concept of innovation for function-parametric processes. In the second part the author summarizes his results from his CWI-Report BS- R890-4 (1989). He assumes that the parametric family of the laws of the random vectors is regular: Fisher information exists and is in a strong sense continuous with respect to the parameter. If the unknown parameter is estimated using a so-called projective estimator, he derives a linear approximation for the transformed empirical process under the null hypothesis and under contiguous alternatives. The key tool is a kind of projection transformation for the empirical process. At the end, the author gives a weak convergence result for the transformed empirical process.

In the last part of the paper the author gives a function parametric version of the earlier results. He shows how one can define innovation processes with respect to a family of orthogonal projections. He then continues by using this family to give some concrete examples, which show how this notion generalizes previous work. This section ends by giving weak convergence results under the null hypothesis and under the contiguous alternatives. The limiting process is distribution free, if the parametric family is regular.

In the first part, the author gives a short history of goodness of fit theory recalling the classical results of Kolmogorov and Doob. He also induces the concept of innovation for function-parametric processes. In the second part the author summarizes his results from his CWI-Report BS- R890-4 (1989). He assumes that the parametric family of the laws of the random vectors is regular: Fisher information exists and is in a strong sense continuous with respect to the parameter. If the unknown parameter is estimated using a so-called projective estimator, he derives a linear approximation for the transformed empirical process under the null hypothesis and under contiguous alternatives. The key tool is a kind of projection transformation for the empirical process. At the end, the author gives a weak convergence result for the transformed empirical process.

In the last part of the paper the author gives a function parametric version of the earlier results. He shows how one can define innovation processes with respect to a family of orthogonal projections. He then continues by using this family to give some concrete examples, which show how this notion generalizes previous work. This section ends by giving weak convergence results under the null hypothesis and under the contiguous alternatives. The limiting process is distribution free, if the parametric family is regular.

Reviewer: E.Valkeila (Helsinki)