Goodness of Fit Problem and Scanning Innovation Martingales

Abstract

This paper is mainly devoted to the following statistical problem: in the case of random variables of any finite dimension and both simple or parametric hypotheses, how to construct convenient "empirical" processes which could provide the basis for goodness of fit tests-more or less in the same way as the uniform empirical process does in the case of simple hypothesis and scalar random variables. The solution of this problem is connected here with the theory of multiparameter martingales and the theory of function-parametric processes. Namely, for the limiting Gaussian processes some kind of filtration is introduced and so-called scanning innovation processes are constructed-the adapted standard Wiener processes in one-to-one correspondence with initial Gaussian processes. This is done for the function-parametric versions of the processes.