Ebook: Gaussian and Non-Gaussian Linear Time Series and Random Fields

Much of this book is concerned with autoregressive and moving av­ erage linear stationary sequences and random fields. These models are part of the classical literature in time series analysis, particularly in the Gaussian case. There is a large literature on probabilistic and statistical aspects of these models-to a great extent in the Gaussian context. In the Gaussian case best predictors are linear and there is an extensive study of the asymptotics of asymptotically optimal esti­ mators. Some discussion of these classical results is given to provide a contrast with what may occur in the non-Gaussian case. There the prediction problem may be nonlinear and problems of estima­ tion can have a certain complexity due to the richer structure that non-Gaussian models may have. Gaussian stationary sequences have a reversible probability struc­ ture, that is, the probability structure with time increasing in the usual manner is the same as that with time reversed. Chapter 1 considers the question of reversibility for linear stationary sequences and gives necessary and sufficient conditions for the reversibility. A neat result of Breidt and Davis on reversibility is presented. A sim­ ple but elegant result of Cheng is also given that specifies conditions for the identifiability of the filter coefficients that specify a linear non-Gaussian random field.

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The book is concerned with linear time series and random fields in both the Gaussian and especially the non-Gaussian context. The principal focus is on autoregressive moving average models and analogous random fields. Probabilistic and statistical questions are both discussed. The Gaussian models are contrasted with noncausal or noninvertible (nonminimum phase) non-Gaussian models which can have a much richer structure than Gaussian models. The book deals with problems of prediction (which can have a nonlinear character) and estimation. New results for nonminimum phase non-Gaussian processes are exposited and open questions are noted. The book is intended as a text for graduate students in statistics, mathematics, engineering, the natural sciences and economics. An initial background in probability theory and statistics is suggested. Notes on background, history and open problems are given at the end of the book. Murray Rosenblatt is Professor of Mathematics at the University of California, San Diego. He was a Guggenheim Fellow in 1965 and 1972 and is a member of the National Academy of Sciences, U.S.A. He is the author of Random Processes (1962), Markov Processes: Structure and Asymptotic Behavior (1971), Stationary Sequences and Random Fields (1985), and Stochastic Curve Estimation (1991).

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The book is concerned with linear time series and random fields in both the Gaussian and especially the non-Gaussian context. The principal focus is on autoregressive moving average models and analogous random fields. Probabilistic and statistical questions are both discussed. The Gaussian models are contrasted with noncausal or noninvertible (nonminimum phase) non-Gaussian models which can have a much richer structure than Gaussian models. The book deals with problems of prediction (which can have a nonlinear character) and estimation. New results for nonminimum phase non-Gaussian processes are exposited and open questions are noted. The book is intended as a text for graduate students in statistics, mathematics, engineering, the natural sciences and economics. An initial background in probability theory and statistics is suggested. Notes on background, history and open problems are given at the end of the book. Murray Rosenblatt is Professor of Mathematics at the University of California, San Diego. He was a Guggenheim Fellow in 1965 and 1972 and is a member of the National Academy of Sciences, U.S.A. He is the author of Random Processes (1962), Markov Processes: Structure and Asymptotic Behavior (1971), Stationary Sequences and Random Fields (1985), and Stochastic Curve Estimation (1991).
Content:
Front Matter....Pages i-xiii
Reversibility and Identifiability....Pages 1-13
Minimum Phase Estimation....Pages 15-26
Homogeneous Gaussian Random Fields....Pages 27-39
Cumulants, Mixing and Estimation for Gaussian Fields....Pages 41-81
Prediction for Minimum and Nonminimum Phase Models....Pages 83-115
The Fluctuation of the Quasi-Gaussian Likelihood....Pages 117-139
Random Fields....Pages 141-154
Estimation for Possibly Nonminimum Phase Schemes....Pages 155-210
Back Matter....Pages 211-246

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The book is concerned with linear time series and random fields in both the Gaussian and especially the non-Gaussian context. The principal focus is on autoregressive moving average models and analogous random fields. Probabilistic and statistical questions are both discussed. The Gaussian models are contrasted with noncausal or noninvertible (nonminimum phase) non-Gaussian models which can have a much richer structure than Gaussian models. The book deals with problems of prediction (which can have a nonlinear character) and estimation. New results for nonminimum phase non-Gaussian processes are exposited and open questions are noted. The book is intended as a text for graduate students in statistics, mathematics, engineering, the natural sciences and economics. An initial background in probability theory and statistics is suggested. Notes on background, history and open problems are given at the end of the book. Murray Rosenblatt is Professor of Mathematics at the University of California, San Diego. He was a Guggenheim Fellow in 1965 and 1972 and is a member of the National Academy of Sciences, U.S.A. He is the author of Random Processes (1962), Markov Processes: Structure and Asymptotic Behavior (1971), Stationary Sequences and Random Fields (1985), and Stochastic Curve Estimation (1991).
Content:
Front Matter....Pages i-xiii
Reversibility and Identifiability....Pages 1-13
Minimum Phase Estimation....Pages 15-26
Homogeneous Gaussian Random Fields....Pages 27-39
Cumulants, Mixing and Estimation for Gaussian Fields....Pages 41-81
Prediction for Minimum and Nonminimum Phase Models....Pages 83-115
The Fluctuation of the Quasi-Gaussian Likelihood....Pages 117-139
Random Fields....Pages 141-154
Estimation for Possibly Nonminimum Phase Schemes....Pages 155-210
Back Matter....Pages 211-246
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