Statistical Inferences for Stochasic Processes: Theory and Methods

Statistical Inference Stochastic Processes provides information pertinent to the theory of stochastic processes. This book discusses stochastic models that are increasingly used in scientific research and describes some of their applications. Organized into three parts encompassing 12 chapters, this book begins with an overview of the basic concepts and procedures of statistical inference. This text then explains the inference problems for Galton–Watson process for discrete time and Markov-branching processes for continuous time. Other chapters consider problems of prediction, filtering, and parameter estimation for some simple discrete-time linear stochastic processes. This book discusses as well the ergodic type chains with finite and countable state-spaces and describes some results on birth and death processes that are of a non-ergodic type. The final chapter deals with inference procedures for stochastic processes through sequential procedures. This book is a valuable resource for graduate students.

Table of Contents:
Preface List of Notation Chapter 0 Introductory Examples of Stochastic Models Example 1. A Random Walk Model for Neuron Firing Example 2. Chain Binomial Models in Epidemiology Example 3. A Population Growth Model Example 4. A Spatial Model for Plant Ecology Example 5. A Cluster Process for Population Settlements Example 6. A Model in Population Genetics Example 7. A Storage Model Example 8. A Compound Poisson Model for Insurance Risk Example 9. System Reliability Models Example 10. A Model for Cell Kinetics Example 11. Queueing Models for Telephone Calls Example 12. Clustering Splitting Model for Animal Behaviour Example 13. Prediction of Economic Time Series Example 14. Signal Estimation Bibliographical Notes Part I Special Models Chapter 1 Basic Principles and Methods of Statistical Inference 1. Introduction 2. The Likelihood Function and Sufficient Statistics 3. Frequency Approach 4. The Bayesian Approach 5. Asymptotic Inference 6. Nonparametric Methods 7. Sequential Methods Bibliographical Notes Chapter 2 Branching Processes 1. Introduction 2. The Galton-Watson Process 3. The Markov Branching Process Bibliographical Notes Complements Chapter 3 Simple Linear Models 1. Introduction 2. Prediction 3. Filtering Problem 4. Parameter Estimation 5. Further Topics Bibliographical Notes Complements Chapter 4 Discrete Markov Chains 1. Introduction 2. Finite Markov Chains 3. A Macro Model (Finite State Space) 4. Grouped Markov Chains (Finite State Space) 5. Countable State Space Bibliographical Notes Complements Chapter 5 Markov Chains in Continuous Time 1. Introduction 2. Finite Markov Chains 3. Queueing Models 4. Pure Birth Process 5. The Birth and Death Process Bibliographical Notes Complements Chapter 6 Simple Point Processes 1. Introduction 2. Homogeneous Poisson Process 3. Non-homogeneous Poisson Process 4. Compound Poisson Process 5. Further Topics Bibliographical Notes Complements Part II General Theory Chapter 7 Large Sample Theory for Discrete Parameter Stochastic Processes 1. Introduction 2. Estimation 3. Efficient Tests of Simple Hypotheses 4. Large Sample Tests 5. Optimal Asymptotic Tests of Composite Hypotheses 6. Further Topics Bibliographical Notes Complements Chapter 8 Large Sample Theory for Continuous Parameter Stochastic Processes 1. Introduction 2. Observable Coordinates 3. The General Problem 4. Testing Hypotheses 5. Estimation 6. Estimating the Infinitesimal Generator for a Continuous Time Finite State Markov Process 7. Further Topics Bibliographical Notes Complements Chapter 9 Diffusion Processes 1. Introduction 2. Diffusion Processes 3. Absolute Continuity of Measures for Diffusion Processes 4. Parameter Estimation in a Linear Stochastic Differential Equation 5. Asymptotic Likelihood Theory for Multidimensional Diffusion Processes 6. Hypotheses Testing for Parameters of Diffusion Processes 7. Sequential Estimation of the Parameters of a Diffusion Process 8. Sequential Test for Diffusion Processes 9. Bayes Estimation for Diffusion Processes 10. Further Topics Bibliographical Notes Complements Part III Further Approaches Chapter 10 Bayesian Inference for Stochastic Processes 1. Introduction 2. Preliminaries 3. The Bernstein-Von Mises Theorem 4. Asymptotic Behaviour of Bayes Estimators 5. Bayesian Testing 6. Further Topics 7. Proof of Tightness of Processes Bibliographical Notes Complements Chapter 11 Nonparametric Inference for Stochastic Processes 1. Introduction 2. Nonparametric Estimation for Stochastic Processes 3. Nonparametric Tests for Stochastic Processes 4. Further Topics Bibliographical Notes Complements Chapter 12 Sequential Inference for Stochastic Processes 1. Introduction 2. Sequential Estimation for Stochastic Processes 3. Sequential Tests of Hypotheses for Stochastic Processes 4. Sequential Tests for the Drift of Wiener Process 5. Further Topics Bibliographical Notes Complements Appendix 1 Martingales 1. Martingales and Limit Theorems 2. A Random Central Limit Theorem for Martingales 3. Embedding Submartingales in Wiener Process with Drift 4. Structure of Continuous Parameter Martingales Appendix 2 Stochastic Differential Equations 1. Stochastic Integrals 2. Central Limit Theorem for Vector-Valued Stochastic Integrals 3. Stochastic Differential Equations Appendix 3 Proof of Sudakov's Lemma (Theorem 2.3) of Chapter 12 Appendix 4 Generalized Functions and Generalized Stochastic Processes References Index