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AuthorRosenblatt, Murray. author
TitleGaussian and Non-Gaussian Linear Time Series and Random Fields [electronic resource] / by Murray Rosenblatt
ImprintNew York, NY : Springer New York : Imprint: Springer, 2000
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Descript XIII, 247 p. online resource


Much of this book is concerned with autoregressive and moving avยญ 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 estiยญ 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 estimaยญ tion can have a certain complexity due to the richer structure that non-Gaussian models may have. Gaussian stationary sequences have a reversible probability strucยญ 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 simยญ 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


1 Reversibility and Identifiability -- 1.1 Linear Sequences and the Gaussian Property -- 1.2 Reversibility -- 1.3 Identifiability -- 1.4 Minimum and Nonminimum Phase Sequences -- 2 Minimum Phase Estimation -- 2.1 The Minimum Phase Case and the Quasi-Gaussian Likelihood -- 2.2 Consistency -- 2.3 The Asymptotic Distribution -- 3 Homogeneous Gaussian Random Fields -- 3.1 Regular and Singular Fields -- 3.2 An Isometry -- 3.3 L-Fields and L-Markov Fields -- 4 Cumulants, Mixing and Estimation for Gaussian Fields -- 4.1 Moments and Cumulants -- 4.2 Higher Order Spectra -- 4.3 Some Simple Inequalities and Strong Mixing -- 4.4 Strong Mixing for Two-Sided Linear Processes -- 4.5 Mixing and a Central Limit Theorem for Random Fields -- 4.6 Estimation for Stationary Random Fields -- 4.7 Cumulants of Finite Fourier Transforms -- 4.8 Appendix: Two Inequalities -- 5 Prediction for Minimum and Nonminimum Phase Models -- 5.1 Introduction -- 5.2 A First Order Autoregressive Model -- 5.3 Nonminimum Phase Autoregressive Models -- 5.4 A Functional Equation -- 5.5 Entropy -- 5.6 Continuous Time Parameter Processes -- 6 The Fluctuation of the Quasi-Gaussian Likelihood -- 6.1 Initial Remarks -- 6.2 Derivation -- 6.3 The Limiting Process -- 7 Random Fields -- 7.1 Introduction -- 7.2 Markov Fields and Chains -- 7.3 Entropy and a Limit Theorem -- 7.4 Some Illustrations -- 8 Estimation for Possibly Nonminimum Phase Schemes -- 8.1 The Likelihood for Possibly Non-Gaussian Autoregressive Schemes -- 8.2 Asymptotic Normality -- 8.3 Preliminary Comments: Approximate Maximum Likelihood Estimates for Non-Gaussian Nonminimum Phase ARMA Sequences -- 8.4 The Likelihood Function -- 8.5 The Covariance Matrix -- 8.6 Solution of the Approximate Likelihood Equations -- 8.7 Cumulants and Estimation for Autoregressive Schemes -- 8.8 Superefficiency -- Bibliographic Notes -- References -- Notation -- Author Index

Mathematics Probabilities Statistics Mathematics Probability Theory and Stochastic Processes Statistical Theory and Methods


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