Possibilities to extend the distribution-free testing of various statistical hypothesis have been among scientific interests of the speaker for relatively long time. The so called Khmaladze transformation, first published in 1981 and 1993, was widely re- searched and became known. The approach that will be presented in this talk is very different, perhaps – simpler, and widely applicable. To illustrate this, we will consider the problems of distribution-free testing for a) parametric families of discrete distributions, b) parametric families of continuous distributions in multidimensional spaces, c) in para- metric regression problems, and to enter the field of stochastic processes, d) for Markov chains and e) for Markov diffusion processes.

Unlike Khmaladze transformation, which was based on the theory of innovation martingales, the current approach uses the group of unitary operators in appropriate functional spaces. We will, so to say, “rotate" various empirical processes into each other. Heuristic motivation behind the approach will be in forefront of the presentation and the technical aspects will be left, mostly, where they are now – in the published papers.​