Mathematical Sciences: Bayesian Modeling and Inference for Time Series with Stable Innovations

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9510348 Ravishanker Abstract The investigator studies Bayesian modeling and inference for time series with infinite variance stable innovations, addressing methodological problems as well as applications to real world data. The justification for the study of infinite variance stable processes stems from empirical evidence for its usefulness in several application areas such as astronomy, economics, engineering, finance and physics. The area of modeling stable processes has several open and challenging problems that are addressed. Available classical methods of estimation and inference do not simultaneously estimate the parameters defining the stable process and the parameters of the time series model. The investigator develops new methodology for modeling data generated by autoregressive fractionally integrated moving average processes with stable innovations that characterize long memory and short memory behavior in 'infinite variance' time series. Inference and prediction are implemented using sampling- based Bayesian techniques through Markov chain Monte Carlo algorithms to generate samples from the target posterior distribution. For stable processes, the form of the likelihood does not, in general, admit a closed analytical form. Hence expressions for the complete conditional distributions of the parameters in terms of the characteristic function of the stable process are combined with the Gibbs sampling algorithm or its variants to generate samples from the required posterior by approximating it by a suitable proposal density. Additionally, the role of algorithms that are useful in generating samples from stable processes directly is studied, incorporating this into the Bayesian framework. Various marginal and joint posterior distributions as well as summary features of these distributions are analyzed, as well as a characterization of predictive distributions that permit model choice and forecasting. The investigator studies modeling and forecasting for time series data assumi ng that the data can take on more extreme values than would usually be the case. There is considerable empirical evidence for this behavior in diverse areas of application such as telecommunications, hydrology, physics, economics and finance and for modeling quantities such as gravitational fields of stars, temperature distributions in nuclear reactors, stresses in crystalline lattices, annual rainfall, stock prices etc. Currently available methods for modeling and forecasting are few and inefficient . The investigator develops new innovative methodology for modeling under the assumption that the data are generated by a popular and useful time series process and by incorporating prior information into the modeling as well.