Universal codes as a basis for nonparametric testing of serial independence for time series

Abstract

We address the problem of nonparametric testing of serial independence for time series and its 9 generalization. More precisely, we consider a stationary and ergodic source p, which generates symbols x 1 . . . x t from some finite set A and a null hypothesis H 0 that p is a Markov source of order at 11 most m, (m 0). The alternative hypothesis H 1 is that the sequence is generated by a stationary and ergodic source, which differs from the source under H 0 . In particular, if m = 0 we have the null hy-13 pothesis H 0 that the sequence is generated by a Bernoulli source (i.e. the hypothesis that x 1 . . . x t are independent). In this paper some new tests that are based on so-called universal codes and universal 15 predictors, are suggested.

Nonparametric Statistical Inference for Ergodic Processes

“…[7,22,1,8,13] and others). Other approaches to statistical analysis of stationary dependent time series include the use of (universal) codes [12,22,21]. Here we first show that distributional distance between stationary ergodic processes can be consistently estimated based on sampling, and then apply it to construct consistent tests for the three problems of statistical analysis described above.…”

“…Other hypothesis testing problems that concern stationary time series include testing for having a certain memory (i.e. testing the hypothesis "k-order Markov process" versus "stationary ergodic, not k-order Markov"), testing for membership to parametric families, and others [12,16,17,21,22]. Some recent general results that characterize those hypotheses about finitely-valued ergodic processes that can be tested are provided in [23].…”

“…On the other hand, the main value of the results is in the demonstration of what is possible in principle; finding practically efficient procedures for each of the considered problems is an interesting problem for further research. A closely related but more practical approach is that of tests based on universal codes [21,20].…”

Testing composite hypotheses about discrete-valued stationary processes

“…For a weaker notion of consistency (the test has to stabilize on the correct answer eventually, with probability 1) the existence of a consistent test for HM k was established in [7]. For the notion of consistency considered here, a consistent test for M k was proposed in [10], while for the case of testing HM k the result above is apparently new.…”

“…The latter condition is that H 0 and H 1 are contained in disjoint F σ sets (countable unions of closed sets), with respect to the topology of weak convergence. For the notion of consistency that we consider, consistent tests for some specific hypotheses, but under the general alternative of stationary ergodic processes, have been proposed in [10], [11], [13], [14], which address problems of testing identity, independence, estimating the order of a Markov process, and also the change point problem. In particular, [13], [14] use tests based on distributional distance.…”