We propose a method for statistical analysis of time series, that allows us to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, we consider three problems: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems we construct a test that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance.

We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply this approach to the following four problems: goodness-of-fit testing (or identity testing), testing for independence, testing of serial independence and homogeneity testing and suggest nonparametric statistical tests for these problems. It is important to note that practically used so-called archivers can be used for suggested testing.

Abstract—In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance. Index Terms—Non-parametric hypothesis testing, stationary ergodic processes, goodness-of-fit test, process classification, change point problem. I.

Two series of binary observations x1, x1,... and y1, y2,... are presented: at each time n ∈ N we are given xn and yn. It is assumed that the sequences are generated independently of each other by two stochastic processes. We are interested in the question of whether the sequences represent a typical realization of two different processes or of the same one. We demonstrate that this is impossible to decide in the case when the processes are B-processes. It follows that discrimination is impossible for the set of all (finite-valued) stationary ergodic processes in general. This result means that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some of such processes. It contrasts earlier positive results on B-processes, in particular those showing that there are consistent ¯ d-distance estimates for this class of processes. Keywords: Process discrimination, B-processes, stationary ergodic processes, time series, homogeneity testing 1

Two series of binary observations x1, x1,... and y1, y2,... are presented: xn and yn are given at each time n ∈ N. It is assumed that the sequences are generated independently of each other by two B-processes. The question of interest is whether the sequences represent a typical realization of two different processes or of the same one. It is demonstrated that this is impossible to decide, in the sense that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some B-processes. This contrasts earlier positive results on B-processes, in particular those showing that there are consistent ¯ d-distance estimates for this class of processes, and on ergodic processes, in particular, those establishing consistent change point estimates. Keywords: Process discrimination, B-processes, stationary ergodic processes, time series, homogeneity testing 1

Abstract Two series of binary observations x1,x1,... and y1,y2,... are presented: xn and yn are given at each time n ∈ N. It is assumed that the sequences are generated independently of each other by two B-processes. The question of interest is whether the sequences represent a typical realization of two different processes or of the same one. It is demonstrated that this is impossible to decide, in the sense that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some B-processes. This contrasts with earlier positive results on B-processes, in particular, those showing that there are consistent ¯d-distance estimates for this class of processes, and on ergodic processes, in particular, those establishing consistent change point estimates. Keywords Process discrimination · B-processes · Stationary ergodic processes ·

Compression-based methods for nonparametric

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