An Introduction to Discrete-Valued Time Series (eBook)

CHRISTIAN H. WEISS is a professor in the Department of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany. His main area of research is discrete-valued time series. He has published numerous articles in this area and given lectures about time series analysis and discrete-valued time series. He has also written five lecture books in German.

Preface xi

About the CompanionWebsite xv

1 Introduction 1

Part I Count Time Series 9

2 A First Approach for Modeling Time Series of Counts: The

Thinning-based INAR(1)Model 11

2.0 Preliminaries: Notation and Characteristics of Count Distributions 11

2.1 The INAR(1) Model for Time-dependent Counts 16

2.1.1 Definition and Basic Properties 17

2.1.2 The Poisson INAR(1) Model 20

2.1.3 INAR(1) Models with More General Innovations 22

2.2 Approaches for Parameter Estimation 26

2.2.1 Method of Moments 26

2.2.2 Maximum Likelihood Estimation 28

2.3 Model Identification 29

2.4 Checking for Model Adequacy 32

2.5 A Real-data Example 34

2.6 Forecasting of INAR(1) Processes 37

3 Further Thinning-based Models for Count Time Series 43

3.1 Higher-order INARMA Models 43

3.2 Alternative Thinning Concepts 54

3.3 The Binomial AR Model 59

3.4 Multivariate INARMA Models 64

4 INGARCH Models for Count Time Series 73

4.1 Poisson Autoregression 73

4.2 Further Types of INGARCH Models 85

4.3 Multivariate INGARCH Models 93

5 Further Models for Count Time Series 95

5.1 Regression Models 95

5.2 Hidden-Markov Models 107

5.3 Discrete ARMA Models 116

Part II Categorical Time Series 119

6 Analyzing Categorical Time Series 121

6.1 Introduction to Categorical Time Series Analysis 122

6.2 Marginal Properties of Categorical Time Series 126

6.3 Serial Dependence of Categorical Time Series 128

7 Models for Categorical Time Series 133

7.1 Parsimoniously Parametrized Markov Models 133

7.2 Discrete ARMA Models 139

7.3 Hidden-Markov Models 146

7.4 Regression Models 151

Part III Monitoring Discrete-Valued Processes 161

8 Control Charts for Count Processes 163

8.1 Introduction to Statistical Process Control 163

8.2 Shewhart Charts for Count Processes 165

8.2.1 Shewhart Charts for i.i.d. Counts 166

8.2.2 Shewhart Charts for Markov-Dependent Counts 171

8.3 Advanced Control Charts for Count Processes 177

8.3.1 CUSUM Charts for i.i.d. Counts 178

8.3.2 CUSUM Charts for Markov-dependent Counts 182

8.3.3 EWMA Charts for Count Processes 186

9 Control Charts for Categorical Processes 193

9.1 Sample-based Monitoring of Categorical Processes 194

9.1.1 Sample-based Monitoring: Binary Case 194

9.1.2 Sample-based Monitoring: Categorical Case 198

9.2 Continuously Monitoring Categorical Processes 203

9.2.1 Continuous Monitoring: Binary Case 203

9.2.2 Continuous Monitoring: Categorical Case 209

Part IV Appendices 213

A Examples of Count Distributions 215

A.1 Count Models for an Infinite Range 215

A.2 Count Models for a Finite Range 221

A.3 Multivariate Count Models 223

B Basics about Stochastic Processes and Time Series 229

B.1 Stochastic Processes: Basic Terms and Concepts 229

B.2 Discrete-Valued Markov Chains 233

B.2.1 Basic Terms and Concepts 233

B.2.2 Stationary Markov Chains 236

B.3 ARMA Models: Definition and Properties 238

B.4 Further Selected Models for Continuous-valued Time Series 243

B.4.1 GARCH Models 243

B.4.2 VARMA Models 245

C Computational Aspects 249

C.1 Some Comments about the Use of R 250

C.2 List of R Codes 253

C.3 List of Datasets 256

References 257

List of Acronyms 275

List of Notations 277

Index 279