Estimation and linear prediction for regression, autoregression and ARMA with infinite variance data - Cline, Daren B. H., author; Resnick, Sidney I., advisor; Brockwell, Peter J., advisor; Locker, John, committee member; Davis, Richard A., committee member; Boes, Duane C., committee member (Colorado State University. Libraries, 1983)
This dissertation is divided into four parts, each of which considers random variables from distributions with regularly varying tails and/or in a stable domain of attraction. Part I considers the existence of infinite series of an independent sequence of such random variables and the relationship of the probability of large values of the series to the probability of large values of the first component. Part II applies Part I in order to provide a linear predictor for ARMA time series (again with regularly varying tails). This predictor is designed to minimize the probability of large prediction errors relative to the tails of the noise distribution. Part III investigates the products of independent random variables where one has distribution in a stable domain of attraction and gives conditions for which the product distribution is in a stable domain of attraction. Part IV considers estimation of the regression parameter in a model where the independent variables are in a stable domain of attraction. Consistency for certain M-estimators is proved. Utilizing portions of Part III this final part gives necessary and sufficient conditions for consistency of least squares estimators and provides the asymptotic distribution of least squares estimators.
Keywords: Random variables; Prediction theory