(SpringerBriefs in Statistics )
M. B. Rajarshi  (auth.)-
Statistical Inference for Discrete Time Stochastic Processes
-Springer India (2013)
ISSN 2191-544X ISSN 2191-5458 (electronic)
ISBN 978-81-322-0762-7 ISBN 978-81-322-0763-4 (eBook)
DOI 10.1007/978-81-322-0763-4
Springer New Delhi Heidelberg New York Dordrecht London
Library of Congress Control Number: 2012949564
? The Author(s) 2012
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Contents
1 CAN Estimators from Dependent Observations. . . . . . . . . . . . . . . 1
1.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Martingales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Mixing Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Empirical Processes of Dependent Observations . . . . . . . . . . . . 9
1.5 CAN Estimation Under Cramér and Other
Regularity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Markov Chains and Their Extensions. . . . . . . . . . . . . . . . . . . . . . 19
2.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Parametric Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Extensions of Markov Chain Models . . . . . . . . . . . . . . . . . . . . 29
2.4 Hidden Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Aggregate Data from Finite Markov Chains . . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Non-Gaussian ARMA Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Integer Valued Non-Negative Auto-Regressive Models . . . . . . . 39
3.2 Auto-Regressive Models for Continuous Random Variables . . . . 41
3.3 Processes Obtained by Minification . . . . . . . . . . . . . . . . . . . . . 44
3.4 Product AR Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 More General Non-Gaussian Sequences . . . . . . . . . . . . . . . . . . 46
3.6 Goodness-of-Fit Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Estimating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Conditional Least Square Estimation . . . . . . . . . . . . . . . . . . . . 55
4.2 Optimal Estimating Functions . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Estimating Functions for Stochastic Models . . . . . . . . . . . . . . . 61
ix
4.4 Estimating Functions for a Vector Parameter . . . . . . . . . . . . . . 65
4.5 Confidence Intervals Based on Estimating Functions . . . . . . . . . 69
4.6 Combining Correlated Estimating Functions . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Estimation of Joint Densities and Conditional Expectation . . . . . . 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Bootstrap and Other Resampling Procedures . . . . . . . . . . . . . . . . 85
6.1 Efron’s Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Markov Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Bootstrap for Stationary and Invertible ARMA Series . . . . . . . . 91
6.5 AR-Sieve Bootstrap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.6 Block-Based Bootstraps for Stationary Sequences . . . . . . . . . . . 95
6.7 Other Block-Based Sample Reuse Methods
for Stationary Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.8 Resampling Based on Estimating Functions . . . . . . . . . . . . . . . 106
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111