Point Processes and Their Statistical Inference

by Alan Karr

Overview

Maintaining the excellent features that made the first edition so popular, this outstanding reference/text presents the only comprehensive treatment of the theory of point processes and statistical inference for point processes-highlighting both pointprocesses on the real line and sp;,.tial point processes. Thoroughly updated and revised to reflect changes since publication of the firstedition, the expanded Second EdiLion now contains a better organized and easierto-understand treatment of stationary point processes ... expanded treatment ofthe multiplicative intensity model ... expanded treatment of survival analysis . ..broadened consideration of applications ... an expanded and extended bibliographywith over 1,000 references ... and more than 3('() end-of-chapter exercises.

Product Details

ISBN-13:	9780367580032
Publisher:	CRC Press
Publication date:	06/30/2020
Series:	Probability: Pure and Applied
Edition description:	2nd ed.
Pages:	512
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Alan F. Karr is Professor of Mathematical Sciences at The Johns Hopkins University, Baltimore, Maryland. The author or coauthor of over 40 research papers, he is a Fellow of the Institute of Mathematical Statistics; a member of the American Mathematical Society, American Statistical Association, Institute of Mathematical Statistics, and Society for Industrial and Applied Mathematics (SIAM), among others; and serves as an Editor for the SIAM Journal on Applied Mathematics and as an Associate Editor for the journals Mathematics of Operations Research Letters and Operations Research Letters. Professor Karr received the B.S. (1969) and M.S. (1970) degrees in industrial engineering and Ph.D. (1973) degree in applied mathematics, all from Northwestern University, Evanston, Illinois.

Table of Contents

Preface to the Second Edition v

Preface to the First Edition vii

1 Point Processes: Distribution Theory 1

1.1 Random Measures and Point Processes 4

1.2 Distributional Descriptors and Uniqueness 8

1.3 Convergence in Distribution 12

1.4 Marked Point Processes; Cluster Processes 16

1.5 Transformations of Point Processes 21

1.6 Approximation of Point Processes 26

1.7 Palm Distributions 31

1.8 Stationary Point Processes 35

Exercises 44

Notes 48

2 Point Processes: Intensity Theory 53

2.1 The History of a Marked Point Process 54

2.2 Stochastic Stieltjes Integration 57

2.3 Compensators 59

2.4 Stochastic Intensities 69

2.5 Representation of Point Process Martingales 73

2.6 Conditioning in General Spaces 76

Exercises 84

Notes 88

3 Inference for Point Processes: An Introduction 93

3.1 Forms of Observation 96

3.1.1 Complete Observation 97

3.1.2 Partial Observation 98

3.2 Statistical Inference 100

3.2.1 Parametric Estimation 104

3.2.2 Nonparametric Estimation 106

3.2.3 Parametric Testing of Quantitative Hypotheses 107

3.2.4 Nonparametric Tests of Quantitative Hypotheses 108

3.2.5 Tests of Qualitative Hypotheses 108

3.3 State Estimation 110

3.4 Combined Inference and State Estimation 114

3.5 Example: Ordinary Poisson Processes 117

3.5.1 Estimation of λ 118

3.5.2 Hypothesis Tests for λ 119

3.5.3 Other Hypothesis Tests 121

3.5.4 State Estimation 122

3.5.5 Combined Inference and State Estimation 122

Exercises 122

Notes 127

4 Empirical Inference for Point Processes 132

4.1 Empirical Measures 133

4.2 Empirical Laplace Functionals 139

4.3 Empirical Zero-Probability Functionals 142

4.4 Estimation of Reduced Palm Distributions 145

4.5 Inference for Thinned Point Processes 149

4.6 Strong Approximation of Poisson Processes 155

Exercises 159

Notes 161

5 Martingale Inference for Point Processes: General Theory 165

5.1 Statistical Models Based on Stochastic Intensities 166

5.2 Asymptotics for Martingale Estimators 174

5.3 The Multiplicative Intensity Model 180

5.3.1 Martingale Estimation 180

5.3.2 Maximum Likelihood Estimation 183

5.3.3 Hypothesis Tests 191

5.4 Filtering with Point Process Observations 194

5.5 The Cox Regression Model 200

5.5.1 The Survival Time Model 201

5.5.2 The Point Process Model 203

5.5.3 Time-Dependent Covariate Effects 208

Exercises 216

Notes 219

6 Inference for Poisson Processes on General Spaces 223

6.1 Estimation 224

6.2 Equivalence and Singularity; Hypothesis Testing 231

6.3 Partially Observed Poisson Processes 238

6.3.1 p-Thinned Poisson Processes 238

6.3.2 Stochastic Integral Process 241

6.3.3 Poisson Processes with One Observable Component 244

6.4 Inference Given Integral Data 247

6.5 Random Measures with Independent Increments; Poisson Cluster Processes 250

Exercises 254

Notes 258

7 Inference for Cox Processes on General Spaces 262

7.1 Estimation 264

7.2 State Estimation 269

7.3 Likelihood Ratios and Hypothesis Tests 281

7.4 Combined Inference and State Estimation 287

7.5 Martingale Inference for Cox Processes 295

Exercises 301

Notes 304

8 Nonparametric Inference for Renewal Processes 307

8.1 Estimation of the Interarrival Distribution 312

8.2 Likelihood Ratios and Hypothesis Tests 319

8.3 State Estimation; Combined Inference and State Estimation 328

8.4 Estimation for Markov Renewal Processes 332

8.4.1 Single Realization Inference 334

8.4.2 Multiple Realization Inference 336

Exercises 340

Notes 344

9 Inference for Stationary Point Processes 348

9.1 Estimation of Palm and Moment Measures 350

9.2 Spectral Analysis 358

9.3 Intensity-Based Inference on R 362

9.4 Linear State Estimation 370

Exercises 376

Notes 379

10 Inference for Stochastic Processes Based on Poisson Process Samples 382

10.1 Markov Processes 384

10.2 Stationary Processes on R 390

10.3 Stationary Random Fields 398

Exercises 407

Notes 409

Appendix A Spaces of Measures 411

Appendix B Continuous Time Martingales 415

Bibliography 423

Index 483