Martingale Methods in Statistics provides a unique introduction to statistics of stochastic processes written with the author’s strong desire to present what is not available in other textbooks. While the author chooses to omit the well-known proofs of some of fundamental theorems in martingale theory by making clear citations instead, the author does his best to describe some intuitive interpretations or concrete usages of such theorems. On the other hand, the exposition of relatively new theorems in asymptotic statistics is presented in a completely self-contained way. Some simple, easy-to-understand proofs of martingale central limit theorems are included.
The potential readers include those who hope to build up mathematical bases to deal with high-frequency data in mathematical finance and those who hope to learn the theoretical background for Cox’s regression model in survival analysis. A highlight of the monograph is Chapters 8-10 dealing with Z-estimators and related topics, such as the asymptotic representation of Z-estimators, the theory of asymptotically optimal inference based on the LAN concept and the unified approach to the change point problems via "Z-process method". Some new inequalities for maxima of finitely many martingales are presented in the Appendix. Readers will find many tips for solving concrete problems in modern statistics of stochastic processes as well as in more fundamental models such as i.i.d. and Markov chain models.
Author(s): Yoichi Nishiyama
Series: Chapman & Hall/CRC Monographs on Statistics and Applied Probability 170
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2021
Language: English
Pages: 260
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Notations and Conventions
List of Figures
I. Introduction
1. Prologue
1.1 Why is the Martingale so Useful?
1.1.1. Martingale as a tool to analyze time series data in real time
1.1.2. Martingale as a tool to deal with censored data correctly
1.2. Invitation to Statistical Modelling with Semimartingales
1.2.1. From non-linear regression to diffusion process model
1.2.2. Cox’s regression model as a semimartingale
2. Preliminaries
2.1. Remarks on Limit Operations in Measure Theory
2.1.1. Limit operations for monotone sequence of measurable sets
2.1.2. Limit theorems for Lebesgue integrals
2.2. Conditional Expectation
2.2.1. Understanding the definition of conditional expectation
2.2.2. Properties of conditional expectation
2.3. Stochastic Convergence
3. A Short Introduction to Statistics of Stochastic Processes
3.1. The “Core” of Statistics
3.1.1. Two illustrations
3.1.2. Filtration, martingale
3.2. A Motivation to Study Stochastic Integrals
3.2.1. Intensity processes of counting processes
3.2.2. Itô integrals and diffusion processes
3.3. Square-Integrable Martingales
3.3.1. Predictable quadratic variations
3.3.2. Stochastic integrals
3.3.3. Introduction to CLT for square-integrable martingales
3.4. Asymptotic Normality of MLEs in Stochastic Process Models
3.4.1. Counting process models
3.4.2. Diffusion process models
3.4.3. Summary of the approach
3.5 Examples
3.5.1. Examples of counting process models
3.5.2. Examples of diffusion process models
II. A User’s Guide to Martingale Methods
4. Discrete-Time Martingales
4.1. Basic Definitions, Prototype for Stochastic Integrals
4.2. Stopping Times, Optional Sampling Theorem
4.3. Inequalities for 1-Dimensional Martingales
4.3.1. Lenglart’s inequality and its corollaries
4.3.2. Bernstein’s inequality
4.3.3. Burkholder’s inequalities
5. Continuous-Time Martingales
5.1. Basic Definitions, Fundamental Facts
5.2. Discre-Time Stochastic Processes in Continuous-Time
5.3. φ (M) Is a Submartingale
5.4. “Predictable” and “Finite-Variation”
5.4.1. Predictable and optional processes
5.4.2. Processes with finite-variation
5.4.3. A role of the two properties
5.5. Stopping Times, First Hitting Times
5.6. Localizing Procedure
5.7. Integrability of Martigales, Optional Sampling Theorem
5.8. Doob-Meyer Decomposition Theorem
5.8.1. Doob’s inequality
5.8.2. Doob-Meyer decomposition theorem
5.9. Predictable Quadratic Co-Variations
5.10. Decompositions of Local Martingales
6. Tools of Semimartingales
6.1. Semimartingales
6.2. Stochastic Integrals
6.2.1. Starting point of constructing stochastic integrals
6.2.2. Stochastic integral w.r.t. locally square-integrable martingale
6.2.3. Stochastic integral w.r.t. semimartingale
6.3. Formula for the Integration by Parts
6.4. Itô’s Formula
6.5. Likelihood Ratio Processes
6.5.1. Likelihood ratio process and martingale
6.5.2. Girsanov’s theorem
6.5.3. Example: Diffusion processes
6.5.4. Example: Counting processes
6.6. Inequalities for 1-Dimensional Martingales
6.6.1. Lenglart’s inequality and its corollaries
6.6.2. Bernstein’s inequality
6.6.3. Burkholder-Davis-Gundy’s inequalities
III. Asymptotic Statistics with Martingale Methods
7. Tools for Asymptotic Statistics
7.1. Martingale Central Limit Theorems
7.1.1. Discrete-time martingales
7.1.2. Continuous local martingales
7.1.3. Stochastic integrals w.r.t. counting processes
7.1.4. Local martingales
7.2. Functional Martingale Central Limit Theorems
7.2.1. Preliminaries
7.2.2. The functional CLT for local martingales
7.2.3. Special cases
7.3. Uniform Convergence of Random Fields
7.3.1. Uniform law of large numbers for ergodic random fields
7.3.2. Uniform convergence of smooth random fields
7.4. Tools for Discrete Sampling of Diffusion Processes
8. Parametric Z-Estimators
8.1. Illustrations with MLEs in I.I.D. Models
8.1.1. Intuitive arguments for consistency of MLEs
8.1.2. Intuitive arguments for asymptotic normality of MLEs
8.2. General Theory for Z-estimators
8.2.1. Consistency of Z-estimators, I
8.2.2. Asymptotic representation of Z-estimators, I
8.3. Examples, I-1 (Fundamental Models)
8.3.1. Rigorous arguments for MLEs in i.i.d. models
8.3.2. MLEs in Markov chain models
8.4 Interim Summary for Approach Overview
8.4.1. Consistency
8.4.2. Asymptotic normality
8.5. Examples, I-2 (Advanced Topics)
8.5.1. Method of moment estimators
8.5.2. Quasi-likelihood for drifts in ergodic diffusion models
8.5.3. Quasi-likelihood for volatilities in ergodic diffusion models
8.5.4. Partial-likelihood for Cox’s regression models
8.6. More General Theory for Z-estimators
8.6.1. Consistency of Z-estimators, II
8.6.2. Asymptotic representation of Z-estimators, II
8.7. Example, II (More Advanced Topic: Different Rates of Convergence)
8.7.1. Quasi-likelihood for ergodic diffusion models
9. Optimal Inference in Finite-Dimensional LAN Models
9.1. Local Asymptotic Normality
9.2. Asymptotic Efficiency
9.3. How to Apply
10. Z-Process Method for Change Point Problems
10.1. Illustrations with Independent Random Sequences
10.2. Z-Process Method: General Theorem
10.3. Examples
10.3.1. Rigorous arguments for independent random sequences
10.3.2. Markov chain models
10.3.3. Final exercises: three models of ergodic diffusions
A. Appendices
A1. Supplements
A1.1. A Stochastic Maximal Inequality and Its Applications
A1.1.1. Continuous-time case
A1.1.2. Discrete-time case
A1.2. Supplementary Tools for the Main Parts
A2. Notes
A3. Solutions/Hints to Exercises
Bibliography
Index
