Bickel, Mathematical Statistics, Basic Ideas And S summer schule madche
- Who are Bickel and Doksum and what are their contributions? - What are the main topics covered in their book series? - What is the purpose and scope of this article? H2: Statistical Models, Goals, and Performance Criteria - What are statistical models and how are they constructed? - What are the goals of statistical inference and how are they related to decision theory? - What are the performance criteria for evaluating statistical methods and how are they measured? - What are some examples of common statistical models and methods? H2: Methods of Estimation - What is estimation and what are its types? - What are the properties of estimators and how are they derived? - What are some examples of classical estimators and their applications? - What are some alternative approaches to estimation, such as Bayesian, robust, and nonparametric methods? H2: Measures of Performances - What are measures of performances and why are they needed? - What are the concepts of bias, variance, mean squared error, efficiency, and consistency? - How can measures of performances be computed and compared? - What are some examples of optimal estimators and their performances? H2: Testing and Confidence Regions: Basic Theory - What is hypothesis testing and what are its types? - What are the concepts of test statistic, critical region, p-value, power, and significance level? - How can tests be constructed and evaluated? - What are some examples of classical tests and their applications? H2: Asymptotic Approximations - What is asymptotic theory and why is it useful? - What are the concepts of convergence in probability, distribution, and mean square? - How can asymptotic approximations be obtained and justified? - What are some examples of asymptotic results and their applications? H2: Inference in the Multiparameter Case - What are the challenges and opportunities of inference in the multiparameter case? - How can estimation, testing, and confidence regions be extended to the multiparameter case? - What are some examples of multiparameter models and methods? - How can dimension reduction techniques be applied to simplify inference in high-dimensional settings? H2: Tools for Asymptotic Analysis - What are some tools for asymptotic analysis and how are they used? - What are the concepts of Taylor expansion, delta method, Slutsky's theorem, Cramér-Rao lower bound, central limit theorem, law of large numbers, etc.? - How can these tools be applied to derive asymptotic approximations for estimators and test statistics? - What are some examples of applications of these tools in various settings? H2: Distribution-Free, Unbiased, And Equivariant Procedures - What are distribution-free, unbiased, and equivariant procedures and why are they desirable? - How can distribution-free procedures be constructed using rank statistics, permutation tests, bootstrap methods, etc.? - How can unbiased procedures be constructed using unbiased estimating equations, minimum variance unbiased estimators, etc.? - How can equivariant procedures be constructed using invariant statistics, group actions, Haar measure, etc.? - What are some examples of applications of these procedures in various settings? H2: Inference in Semiparametric Models - What are semiparametric models and why are they useful? - How can inference be performed in semiparametric models using likelihood-based methods, estimating equations methods, empirical likelihood methods, etc.? - How can efficiency bounds be derived for semiparametric models using influence functions, tangent spaces, etc.? - What are some examples of applications of semiparametric models and methods in various settings? H2: Monte Carlo Methods - What are Monte Carlo methods and why are they needed? - How can Monte Carlo methods be used to generate random samples, evaluate integrals, approximate distributions, etc.? - What are some techniques to improve the accuracy and efficiency of Monte Carlo methods, such as variance reduction, importance sampling, Markov chain Monte Carlo, etc.? - What are some examples of applications of Monte Carlo methods in various settings? H2: Nonparametric Inference for Functions of One Variable - What are nonparametric models and why are they flexible? - How can nonparametric inference be performed for functions of one variable using kernel methods, spline methods, wavelet methods, etc.? - How can the performance of nonparametric methods be assessed using bias-variance trade-off, bandwidth selection, cross-validation, etc.? - What are some examples of applications of nonparametric methods for functions of one variable in various settings? H2: Prediction and Machine Learning - What is prediction and how is it different from inference? - What are the goals and challenges of prediction and machine learning? - How can prediction and machine learning be performed using linear models, nonlinear models, tree-based models, neural networks, etc.? - How can the performance of prediction and machine learning methods be assessed using loss functions, accuracy measures, overfitting, regularization, etc.? - What are some examples of applications of prediction and machine learning methods in various settings? H1: Conclusion - What are the main takeaways from this article? - How can the readers learn more about mathematical statistics and the topics covered in this article? - What are some open problems and future directions for research in mathematical statistics? H1: FAQs - Q1: What is the difference between mathematical statistics and applied statistics? - Q2: What are some prerequisites for studying mathematical statistics at the doctorate level? - Q3: What are some other books or resources on mathematical statistics that complement Bickel and Doksum's book series? - Q4: What are some applications of mathematical statistics in real-world problems or domains? - Q5: How can I practice or improve my skills in mathematical statistics? Table 2: Article with HTML formatting Introduction
Mathematical statistics is a branch of mathematics that deals with the theory and methods of collecting, analyzing, and interpreting data. It provides the foundation for statistical inference, which is the process of drawing conclusions from data. Mathematical statistics also explores the properties and limitations of statistical procedures, such as estimation, testing, confidence intervals, prediction, and decision making.
Mathematical statistics is a rich and diverse field that has many connections with other areas of mathematics, such as probability theory, analysis, algebra, geometry, optimization, etc. It also has many applications in various disciplines, such as natural sciences, engineering, social sciences, medicine, economics, etc.
One of the most influential and comprehensive books on mathematical statistics is the two-volume series by Peter J. Bickel and Kjell A. Doksum. Bickel is a professor emeritus of statistics and mathematics at the University of California, Berkeley. He has made significant contributions to various topics in mathematical statistics, such as asymptotic theory, semiparametric models, robustness, bootstrap methods, etc. Doksum is a professor emeritus of statistics at the University of Wisconsin-Madison. He has also made important contributions to various topics in mathematical statistics, such as rank tests, nonparametric inference, survival analysis, etc.
The book series by Bickel and Doksum covers fundamental and advanced topics in mathematical statistics at the doctorate level. The first volume presents basic ideas and selected topics in classical statistical theory without using measure theory. It covers topics such as statistical models, goals, performance criteria; methods of estimation; measures of performances; testing and confidence regions; asymptotic approximations; inference in the multiparameter case; etc. The second volume covers more modern topics that are important in current research and practice. It covers topics such as tools for asymptotic analysis; distribution-free procedures; inference in semiparametric models; Monte Carlo methods; nonparametric inference for functions of one variable; prediction and machine learning; etc.
The purpose of this article is to provide an overview and summary of the main topics covered in Bickel and Doksum's book series on mathematical statistics. We will also provide some examples and applications to illustrate the concepts and methods discussed in the book series. We hope that this article will serve as a useful