# Statistical Inferences for Stochasic Processes

## 1st Edition

### Theory and Methods

**Authors:**Ishwar V. Basawa

**eBook ISBN:**9781483296142

**Imprint:**Academic Press

**Published Date:**28th January 1980

**Page Count:**435

## Description

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

## Details

- No. of pages:
- 435

- Language:
- English

- Copyright:
- © Academic Press 1980

- Published:
- 28th January 1980

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483296142