Limit Theorems and Statistical Inference for Ergodic Processes

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Abstract 0102268Limit Theorems and Statistical Inference for Ergodic ProcessesA major goal of the project is to develop a new approach to the change point problem in which the abrupt change of the latter is replaced by an arbitrary monotonic change. The new procedure uses a penalized likelihood ratio statistic for testing equality of means against a non-decreasing trend, derived for independent normal observation errors. The properties of the test can be studied in the more general context of dependent, but stationary and ergodic errors. Applications of such procedures should be evident in the analysis of climate changes results from cataclysmic events or legal intervention, such as the required reduction on vehicle emissions.Current work by the principal investigator and students has determined the asymptotic null distribution of the test statistic for stationary ergodic errors under modest conditions, thus allowing application to historical data sets, like weather data. Remaining questions include developing a sequential analogue for applications to quality control, alternative penalizations, and estimating a variance parameter after an isotonic regression. A second major goal of the project is to develop asymptotic distribution theory in a context that is applicable to the first. The central limit theorem will be studied for additive functionals of a Markov chain with special attention to chains in which the current state is a function of the previous state and an independent variable. Many linear and non-linear time series models are of this form. Conditions for the existence of a stationary distribution have been widely studied for such processes, but there is much less work on central limit theory for their additive functionals. The principal investigator plans to develop central limit theory in this context. Previous work has shown that additive functionals can be written as a martingale plus a remainder term of smaller order in many cases, and then asymptotic normality can be deduced from the martingale central limit theorem. This approach does not require Harris recurrence or other strong forms of asymptotic independence. It will be developed, and statistical applications explored, especially applications to the modified change point problem. Other statistical applications include setting approximate confidence intervals. In some cases, approximate confidence intervals may be obtained from a multivariate central limit theorem. For others, it is necessary to develop tightness of empirical processes, and this question will be studied. In highly structured models, it is possible to go beyond asymptotic normality to (Edgeworth like) asymptotic expansions from which corrected confidence intervals can be formed, intervals whose actual coverage probability converges to the nominal value at a fast rate. A third major objective of project is to develop such expansions. Previous work by the principal investigator, co-workers, and students has developed expansions of this nature for adaptively designed linear models and auto regressive processes. This work will be extended to processes whose finite dimensional distributions form exponential families, a large class of processes that includes Markov Chains and many semi-Markov processes.