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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
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.
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.
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 master's 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.
Monographs on statistics and applied probability (Series) ; 170