Asymptotic Statistics With a View to Stochastic Processes 1st Edition by Reinhard Höpfner ISBN 3110250241 978-3110250244

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ISBN 10: 3110250241

ISBN 13: 978-3110250244

Author: Reinhard Höpfner

This textbook is devoted to the general asymptotic theory of statistical experiments. Local asymptotics for statistical models in the sense of local asymptotic (mixed) normality or local asymptotic quadraticity make up the core of the book. Numerous examples deal with classical independent and identically distributed models and with stochastic processes.

The book can be read in different ways, according to possibly different mathematical preferences of the reader. One reader may focus on the statistical theory, and thus on the chapters about Gaussian shift models, mixed normal and quadratic models, and on local asymptotics where the limit model is a Gaussian shift or a mixed normal or a quadratic experiment (LAN, LAMN, LAQ). Another reader may prefer an introduction to stochastic process models where given statistical results apply, and thus concentrate on subsections or chapters on likelihood ratio processes and some diffusion type models where LAN, LAMN or LAQ occurs. Finally, readers might put together both aspects.

The book is suitable for graduate students starting to work in statistics of stochastic processes, as well as for researchers interested in a precise introduction to this area.

Asymptotic Statistics With a View to Stochastic Processes 1st Table of contents: 

• Introduction

  • Overview of Asymptotic Statistics
  • The Importance of Asymptotic Behavior in Statistical Inference
  • Stochastic Processes and Their Role in Statistics
  • Objectives of the Book

• Chapter 1: Foundations of Probability and Stochastic Processes

  • Review of Probability Theory
  • Key Concepts in Stochastic Processes (e.g., Markov Chains, Brownian Motion)
  • Distribution Functions and Their Asymptotics
  • Convergence Types: Weak, Strong, and Distributional Convergence

• Chapter 2: Asymptotics in Statistical Inference

  • The Role of Large Sample Theory in Statistics
  • Consistency of Estimators
  • Asymptotic Normality and the Central Limit Theorem
  • Efficiency and the Cramer-Rao Bound

• Chapter 3: Large Sample Behavior of Estimators

  • Asymptotic Properties of Maximum Likelihood Estimators
  • Asymptotic Variance and Bias of Estimators
  • Law of Large Numbers and its Implications for Estimation
  • The Role of Empirical Processes in Large Sample Theory

• Chapter 4: Stochastic Processes in Statistical Models

  • Introduction to Stochastic Processes in Statistics
  • Stationary and Non-Stationary Processes
  • Markov Chains and Their Applications in Statistics
  • Brownian Motion and Its Role in Asymptotic Statistics

• Chapter 5: Asymptotic Theory for Stochastic Processes

  • Large Sample Theory for Stochastic Processes
  • Asymptotics of Estimators in Stochastic Process Models
  • Central Limit Theorem for Stochastic Processes
  • Convergence of Random Walks and Their Applications

• Chapter 6: Asymptotic Methods in Hypothesis Testing

  • Asymptotic Behavior of Test Statistics
  • Likelihood Ratio Tests and Their Asymptotics
  • Nonparametric Testing and Asymptotic Efficiency
  • Power and Size of Asymptotic Tests

• Chapter 7: The Role of Resampling Techniques in Asymptotic Statistics

  • Bootstrapping and its Asymptotic Behavior
  • Jackknife and Cross-Validation Methods
  • Comparison of Resampling and Asymptotic Methods

• Chapter 8: Advanced Topics in Asymptotic Statistics

  • Asymptotic Methods for Non-Independent Data
  • Long-Range Dependence and Its Statistical Implications
  • Semi-Parametric Models and Asymptotic Theory
  • Asymptotics in High Dimensions

• Chapter 9: Applications of Asymptotic Statistics to Real-World Problems

  • Case Studies in Financial Modeling and Risk Assessment
  • Asymptotic Methods in Time Series Analysis
  • Applications in Queuing Theory and Reliability Engineering

• Chapter 10: Future Directions in Asymptotic and Stochastic Process Statistics

  • Recent Advances in Asymptotic Theory
  • Challenges in High-Dimensional Statistics
  • The Role of Computational Methods in Asymptotic Analysis

• Conclusion

  • Summary of Key Concepts in Asymptotic Statistics and Stochastic Processes
  • Open Problems and Areas for Further Research
  • Final Thoughts on the Future of Asymptotic Statistics

• Appendices

  • Appendix A: Mathematical Preliminaries
  • Appendix B: Table of Common Asymptotic Distributions
  • Appendix C: Supplementary Proofs and Theorems

• References

  • A comprehensive list of references and further readings on Asymptotic Statistics and Stochastic Processes

• Index

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Reinhard Höpfner,Asymptotic Statistics,Stochastic Processes