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E-raamat: Martingale Methods in Statistics

(School of International Liberal Studies, Faculty of International Research and Education, Waseda University)
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Martingale Methods in Statistics provides a unique introduction to statistics of stochastic processes written with the authors 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 Coxs 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.

Arvustused

"This book is expected to be an excellent reference for researchers who need to perform statistical analysis based on the martingale theory. It is very rare to find a book that systematically and rigorously summarizes martingale theory from the point of view of its application to statistics. This textbook harmonically organizes the mathematical facts related to martingales and their statistical applications, and by doing so, it helps researchers to establish a theoretically concrete foundation. Therefore, this book can be evaluated as an excellent textbook where mathematics and statistics meet together." -Insuk Seo, in Journal of the American Statistical Association, November 2023

"The martingale theory is an important topic in probability theory and related tools have been widely applied in statistical analysis, such as financial data or survival analysis. ...This book well summarizes useful tools in martingale and provides rigorous theorems. ... In summary, this book is a nice reference because of rich and comprehensive materials in martingale theory. This book is suitable to researchers who are working on related research topics." - Li-Pang Chen, in Journal of the Royal Statistical Society: Series A, April 2022

Preface ix
Notations and Conventions xi
List of Figures
xiii
I Introduction
1(50)
1 Prologue
3(10)
1.1 Why is the Martingale so Useful?
3(4)
1.1.1 Martingale as a tool to analyze time series data in real time
3(2)
1.1.2 Martingale as a tool to deal with censored data correctly
5(2)
1.2 Invitation to Statistical Modelling with Semimartingales
7(6)
1.2.1 From non-linear regression to diffusion process model
7(2)
1.2.2 Cox's regression model as a semimartingale
9(4)
2 Preliminaries
13(14)
2.1 Remarks on Limit Operations in Measure Theory
13(4)
2.1.1 Limit operations for monotone sequence of measurable sets
13(1)
2.1.2 Limit theorems for Lebesgue integrals
14(3)
2.2 Conditional Expectation
17(3)
2.2.1 Understanding the definition of conditional expectation
17(2)
2.2.2 Properties of conditional expectation
19(1)
2.3 Stochastic Convergence
20(7)
3 A Short Introduction to Statistics of Stochastic Processes
27(24)
3.1 The "Core" of Statistics
27(4)
3.1.1 Two illustrations
27(3)
3.1.2 Filtration, martingale
30(1)
3.2 A Motivation to Study Stochastic Integrals
31(3)
3.2.1 Intensity processes of counting processes
31(2)
3.2.2 Ito integrals and diffusion processes
33(1)
3.3 Square-Integrable Martingales
34(3)
3.3.1 Predictable quadratic variations
34(1)
3.3.2 Stochastic integrals
35(1)
3.3.3 Introduction to CLT for square-integrable martingales
36(1)
3.4 Asymptotic Normality of MLFs in Stochastic Process Models
37(5)
3.4.1 Counting process models
37(2)
3.4.2 Diffusion process models
39(2)
3.4.3 Summary of the approach
41(1)
3.5 Examples
42(9)
3.5.1 Examples of counting process models
42(6)
3.5.2 Examples of diffusion process models
48(3)
II A User's Guide to Martingale Methods
51(66)
4 Discrete-Time Martingales
53(10)
4.1 Basic Definitions, Prototype for Stochastic Integrals
53(3)
4.2 Stopping Times, Optional Sampling Theorem
56(1)
4.3 Inequalities for 1-Dimensional Martingales
57(6)
4.3.1 Lenglart's inequality and its corollaries
57(3)
4.3.2 Bernstein's inequality
60(1)
4.3.3 Burkholder's inequalities
60(3)
5 Continuous-Time Martingales
63(30)
5.1 Basic Definitions, Fundamental Facts
63(5)
5.2 Discre-Time Stochastic Processes in Continuous-Time
68(1)
5.3 φ(M) Is a Submartingale
69(1)
5.4 "Predictable" and "Finite-Variation"
70(4)
5.4.1 Predictable and optional processes
70(1)
5.4.2 Processes with finite-variation
71(3)
5.4.3 A role of the two properties
74(1)
5.5 Stopping Times, First Hitting Times
74(3)
5.6 Localizing Procedure
77(2)
5.7 Integrability of Martigales, Optional Sampling Theorem
79(5)
5.8 Doob-Meyer Decomposition Theorem
84(4)
5.8.1 Doob's inequality
84(1)
5.8.2 Doob-Meyer decomposition theorem
85(3)
5.9 Predictable Quadratic Co-Variations
88(2)
5.10 Decompositions of Local Martingales
90(3)
6 Tools of Semimartingales
93(24)
6.1 Semimartingales
93(2)
6.2 Stochastic Integrals
95(5)
6.2.1 Starting point of constructing stochastic integrals
95(1)
6.2.2 Stochastic integral W.R.T. locally square-integrable martingale
96(2)
6.2.3 Stochastic integral W.R.T. semimartingale
98(2)
6.3 Formula for the Integration by Parts
100(1)
6.4 Ito's Formula
101(3)
6.5 Likelihood Ratio Processes
104(8)
6.5.1 Likelihood ratio process and martingale
104(2)
6.5.2 Girsanov's theorem
106(4)
6.5.3 Example: Diffusion processes
110(1)
6.5.4 Example: Counting processes
111(1)
6.6 Inequalities for 1-Dimensional Martingales
112(5)
6.6.1 Lenglart's inequality and its corollaries
112(3)
6.6.2 Bernstein's inequality
115(1)
6.6.3 Burkholder-Davis-Gundy's inequalities
115(2)
III Asymptotic Statistics with Martingale Methods
117(92)
7 Tools for Asymptotic Statistics
119(36)
7.1 Martingale Central Limit Theorems
119(16)
7.1.1 Discrete-time martingales
119(6)
7.1.2 Continuous local martingales
125(2)
7.1.3 Stochastic integrals W.R.T. counting processes
127(3)
7.1.4 Local martingales
130(5)
7.2 Functional Martingale Central Limit Theorems
135(8)
7.2.1 Preliminaries
136(1)
7.2.2 The functional CLT for local martingales
137(3)
7.2.3 Special cases
140(3)
7.3 Uniform Convergence of Random Fields
143(7)
7.3.1 Uniform law of large numbers for ergodic random fields
143(5)
7.3.2 Uniform convergence of smooth random fields
148(2)
7.4 Tools for Discrete Sampling of Diffusion Processes
150(5)
8 Parametric Z-Estimators
155(38)
8.1 Illustrations with MLEs in I.I.D. Models
155(4)
8.1.1 Intuitive arguments for consistency of MLEs
157(1)
8.1.2 Intuitive arguments for asymptotic normality of MLEs
158(1)
8.2 General Theory for Z-estimators
159(4)
8.2.1 Consistency of Z-estimators, I
160(1)
8.2.2 Asymptotic representation of Z-estimators, I
161(2)
8.3 Examples, I-1 (Fundamental Models)
163(7)
8.3.1 Rigorous arguments for MLEs in I.I.D. models
163(2)
8.3.2 MLEs in Markov chain models
165(5)
8.4 Interim Summary for Approach Overview
170(1)
8.4.1 Consistency
170(1)
8.4.2 Asymptotic normality
171(1)
8.5 Examples, 1-2 (Advanced Topics)
171(17)
8.5.1 Method of moment estimators
171(1)
8.5.2 Quasi-likelihood for drifts in ergodic diffusion models
172(6)
8.5.3 Quasi-likelihood for volatilities in ergodic diffusion models
178(6)
8.5.4 Partial-likelihood for Cox's regression models
184(4)
8.6 More General Theory for Z-estimators
188(2)
8.6.1 Consistency of Z-estimators, II
189(1)
8.6.2 Asymptotic representation of Z-estimators, II
189(1)
8.7 Example, II (More Advanced Topic: Different Rates of Convergence)
190(3)
8.7.1 Quasi-likelihood for ergodic diffusion models
190(3)
9 Optimal Inference in Finite-Dimensional LAN Models
193(4)
9.1 Local Asymptotic Normality
193(1)
9.2 Asymptotic Efficiency
194(1)
9.3 How to Apply
195(2)
10 Z-Process Method for Change Point Problems
197(12)
10.1 Illustrations with Independent Random Sequences
197(2)
10.2 Z-Process Method: General Theorem
199(3)
10.3 Examples
202(7)
10.3.1 Rigorous arguments for independent random sequences
202(1)
10.3.2 Markov chain models
203(3)
10.3.3 Final exercises: three models of ergodic diffusions
206(3)
A Appendices
209(30)
A1 Supplements
211(14)
A1.1 A Stochastic Maximal Inequality and Its Applications
211(1)
A1.1.1 Continuous-time case
212(6)
A1.1.2 Discrete-time case
218(4)
A1.2 Supplementary Tools for the Main Parts
222(3)
A2 Notes
225(4)
A3 Solutions/Hints to Exercises
229(10)
Bibliography 239(4)
Index 243
Yoichi Nishiyama is a professor in mathematical statistics and probability at the School of International Liberal Studies of Waseda University; he is also engaged in the education of masters and doctoral students at the Department of Pure and Applied Mathematics at the same university. Prior to his assignment to Waseda University, he worked at the Institute of Statistical Mathematics, Tokyo, from 1994 to 2015. He was the Editor-in-Chief of Journal of the Japan Statistical Society and a Co-Editor of Annals of the Institute of Statistical Mathematics and he received the JSS Ogawa Award from the Japan Statistical Society in 2009.
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