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Department of Statistics

Permanent URI for this communityhttps://hdl.handle.net/10217/100518

These digital collections include theses, dissertations, and datasets from the Department of Statistics. Due to departmental name changes, materials from the following historical department are also included here: Mathematics and Statistics.

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Browsing Department of Statistics by Author "Brockwell, Peter J., advisor"

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    Estimation and linear prediction for regression, autoregression and ARMA with infinite variance data
    (Colorado State University. Libraries, 1983) 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
    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.
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    Estimation for Lévy-driven CARMA processes
    (Colorado State University. Libraries, 2008) Yang, Yu, author; Brockwell, Peter J., advisor; Davis, Richard A., advisor
    This thesis explores parameter estimation for Lévy-driven continuous-time autoregressive moving average (CARMA) processes, using uniformly and closely spaced discrete-time observations. Specifically, we focus on developing estimation techniques and asymptotic properties of the estimators for three particular families of Lévy-driven CARMA processes. Estimation for the first family, Gaussian autoregressive processes, was developed by deriving exact conditional maximum likelihood estimators of the parameters under the assumption that the process is observed continuously. The resulting estimates are expressed in terms of stochastic integrals which are then approximated using the available closely-spaced discrete-time observations. We apply the results to both linear and non-linear autoregressive processes. For the second family, non-negative Lévy-driven Ornestein-Uhlenbeck processes, we take advantage of the non-negativity of the increments of the driving Lévy process to derive a highly efficient estimation procedure for the autoregressive coefficient when observations are available at uniformly spaced times. Asymptotic properties of the estimator are also studied and a procedure for obtaining estimates of the increments of the driving Lévy process is developed. These estimated increments are important for identifying the nature of the driving Lévy process and for estimating its parameters. For the third family, non-negative Lévy-driven CARMA processes, we estimate the coefficients by maximizing the Gaussian likelihood of the observations and discuss the asymptotic properties of the estimators. We again show how to estimate the increments of the background driving Lévy process and hence to estimate the parameters of the Lévy process itself. We assess the performance of our estimation procedures by simulations and use them to fit models to real data sets in order to determine how the theory applies in practice.
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    Statistical modeling with COGARCH(p,q) processes
    (Colorado State University. Libraries, 2009) Chadraa, Erdenebaatar, author; Brockwell, Peter J., advisor
    In this paper, a family of continuous time GARCH processes, generalizing the COGARCH(1, 1) process of Klüppelberg, et al. (2004), is introduced and studied. The resulting COGARCH(p,q) processes, q ≥ p ≥ 1, exhibit many of the characteristic features of observed financial time series, while their corresponding volatility and squared increment processes display a broader range of autocorrelation structures than those of the COGARCH(1, 1) process. We establish sufficient conditions for the existence of a strictly stationary non-negative solution of the equations for the volatility process and, under conditions which ensure the finiteness of the required moments, determine the autocorrelation functions of both the volatility and squared increment processes. The volatility process is found to have the autocorrelation function of a continuous-time ARMA process while the squared increment process has the autocorrelation function of an ARMA process.