Testing composite hypotheses about discrete-valued stationary processes

IEEE Trans. Inform. Theory

2010

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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.

Section: Introductionmentioning

confidence: 99%

Nonparametric Statistical Inference for Ergodic Processes

IEEE Trans. Inform. Theory

2010

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“…Prior work. This work continuous our previous research [13,14], which provides similar necessary and sufficient conditions for the existence of a consistent test, for a weaker notion of asymmetric consistency: Type I error is uniformly bounded, while Type II error is required to tend to 0 as the sample size grows.…”

Section: Introductionmentioning

confidence: 56%

Uniform hypothesis testing for finite-valued stationary processes

2012

Statistics

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Given a discrete-valued sample X1, . . . , Xn we wish to decide whether it was generated by a distribution belonging to a family H0, or it was generated by a distribution belonging to a family H1. In this work we assume that all distributions are stationary ergodic, and do not make any further assumptions (e.g. no independence or mixing rate assumptions). We would like to have a test whose probability of error (both Type I and Type II) is uniformly bounded. More precisely, we require that for each ε there exist a sample size n such that probability of error is upper-bounded by ε for samples longer than n. We find some necessary and some sufficient conditions on H0 and H1 under which a consistent test (with this notion of consistency) exists. These conditions are topological, with respect to the topology of distributional distance.

“…It is worth noting that the information theory is deeply connected with statistics of time series and signal processing; see, for example, [1,2,5,7,10,11,13,14,15] and [8,9,6], correspondingly. In this paper a new connection of this kind is established: it is shown that the Shannon entropy determines the rate of growth of the size of the confidence set for the signal, given its version corrupted by stationary noise.…”

Section: Introductionmentioning

confidence: 99%

Confidence sets in time-series filtering

2011 IEEE International Symposium on Information Theory Proceedings

2011

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Abstract-The problem of filtering of finite-alphabet stationary ergodic time series is considered. A method for constructing a confidence set for the (unknown) signal is proposed, such that the resulting set has the following properties: First, it includes the unknown signal with probability γ, where γ is a parameter supplied to the filter. Second, the size of the confidence sets grows exponentially with the rate that is asymptotically equal to the conditional entropy of the signal given the data. Moreover, it is shown that this rate is optimal.