Contributions on Theory of Mathematical Statistics

✍ Scribed by Kei Takeuchi

• Publisher: Springer
• Tongue: English
• Leaves: 428
• Category: Library

✦ Synopsis

This volume is a reorganized edition of Kei Takeuchi’s works on various problems in mathematical statistics based on papers and monographs written since the 1960s on several topics in mathematical statistics and published in various journals in English and in Japanese. They are organized into seven parts, each of which is concerned with specific topics and edited to make a consistent thesis. Sometimes expository chapters have been added. The topics included are as follows: theory of statistical prediction from a non-Bayesian viewpoint and analogous to the classical theory of statistical inference; theory of robust estimation, concepts, and procedures, and its implications for practical applications; theory of location and scale covariant/invariant estimations with derivation of explicit forms in various cases; theory of selection and testing of parametric models and a comprehensive approach including the derivation of the Akaike’s Information Criterion (AIC); theory of randomized designs, comparisons of random and conditional approaches, and of randomized and non-randomized designs, with random sampling from finite populations considered as a special case of randomized designs and with some separate independent papers included. Theory of asymptotically optimal and higher-order optimal estimators are not included, since most of them already have been published in the Joint Collected Papers of M. Akahira and K. Takeuchi. There are some topics that are not necessarily new, do not seem to have attracted many theoretical statisticians, and do not appear to have been systematically dealt with in textbooks or expository monographs. One purpose of this volume is to give a comprehensive view of such problems as well.

✦ Table of Contents

Preface
Contents
Part I Statistical Prediction
1 Theory of Statistical Prediction
1.1 Introduction
1.2 Sufficiency with Respect to Prediction
1.3 Point Prediction
1.4 Interval or Region Prediction
1.5 Non-parametric Prediction Regions
1.6 Dichotomous Prediction
1.7 Multiple Prediction
References
Part II Unbiased Estimation
2 Unbiased Estimation in Case of the Class of Distributions of Finite Rank
2.1 Definitions
2.2 Minimum Variance Unbiased Estimators
2.3 Example
2.4 Non-regular Cases
References
3 Some Theorems on Invariant Estimators of Location
3.1 Introduction
3.2 Estimation of the Location Parameter When the Scale is Known
3.3 Some Examples: Scale Known
3.4 Estimation of the Location Parameter When the Scale is Unknown
3.5 Some Examples: Scale Unknown
3.6 Estimation of Linear Regression Coefficients
References
Part III Robust Estimation
4 Robust Estimation and Robust Parameter
4.1 Introduction
4.2 Definition of Location and Scale Parameters
4.3 The Optimum Definition of Location Parameter
4.4 Robust Estimation of Location Parameter
4.5 Definition of the Parameter Depending on Several Distributions
4.6 Construction of Uniformly Efficient Estimator
References
5 Robust Estimation of Location in the Case of Measurement of Physical Quantity
5.1 Introduction
5.2 Nature of Assumptions
5.3 Normative Property of the Normal Distribution
5.4 Class of Asymptotically Efficient Estimators
5.5 Linear Estimators
5.6 Class of M Estimators
5.7 Estimators Derived from Non-parametric Tests
5.8 Conclusions
References
6 A Uniformly Asymptotically Efficient Estimator of a Location Parameter
6.1 Introduction
6.2 The Method
6.3 Monte Carlo Experiments
6.4 Observations on Monte Carlo Results
References
Part IV Randomization
7 Theory of Randomized Designs
7.1 Introduction
7.2 The Model
7.3 Testing the Hypothesis in Randomized Design
7.4 Considerations of the Power of the Tests
References
8 Some Remarks on General Theory for Unbiased Estimation of a Real Parameter of a Finite Population
8.1 Formulation of the Problem
8.2 Estimability
8.3 Ω0-exact Estimators
8.4 Linear Estimators
8.5 Invariance
References
Part V Tests of Normality
9 The Studentized Empirical Characteristic Function and Its Application to Test for the Shape of Distribution
9.1 Introduction
9.2 Limiting Processes
9.3 Application to Test for Normality
9.4 Asymptotic Consideration on the Power
9.4.1 The Power of b2 b2 b2 b2, an(t) an(t) an(t) an(t), tildea a a an n n n(t t t t)
9.4.2 Relative Efficiency
9.5 Moments
9.6 Empirical Study of Power
9.6.1 Null Percentiles of an(t) an(t) an(t) an(t) and tildea a a an n n n(t t t t)
9.6.2 Details of the Simulation
9.6.3 Results and Observations
9.7 Concluding Remarks
References
10 Tests of Univariate Normality
10.1 Introduction
10.2 Tests Based on the Chi-Square Goodness of Fit Type
10.3 Asymptotic Powers of the χ2-type Tests
10.4 Tests Based on the Empirical Distribution
10.5 Tests Based on the Transformed Variables
10.6 Tests Based on the Characteristics of the Normal Distribution
References
11 The Tests for Multivariate Normality
11.1 Basic Properties of the Studentized Multivariate Variables
11.2 Tests of Multivariate Normality
11.3 Tests Based on the Third-Order Cumulants
References
Part VI Model Selection
12 On the Problem of Model Selection Based on the Data
12.1 Fisher's Formulation
12.2 Search for Appropriate Models
12.3 Construction of Models
12.4 Selection of the Model
12.5 More General Approach
12.6 Derivation of AIC
12.7 Problems of AIC
12.8 Some Examples
12.9 Some Additional Remarks
References
Part VII Asymptotic Approximation
13 On Sum of 0–1 Random Variables I. Univariate Case
13.1 Introduction
13.2 Notations and Definitions
13.3 Approximation by Binomial Distribution
13.4 Convergence to Poisson Distribution
13.5 Convergence to the Normal Distribution
References
14 On Sum of 0–1 Random Variables II. Multivariate Case
14.1 Introduction
14.2 Sum of Vectors of 0–1 Random Variables
14.2.1 Notations and Definitions
14.2.2 Approximation by Binomial Distribution
14.2.3 Convergence to Poisson Distribution
14.2.4 Convergence to the Normal Distribution
14.3 Sum of Multinomial Random Vectors
14.3.1 Notations and Definitions
14.3.2 Generalized Krawtchouk Polynomials and Approximation by Multinomial Distribution
14.3.3 Convergence to Poisson Distribution
14.3.4 Convergence to the Normal Distribution
References
15 Algebraic Properties and Validity of Univariate and Multivariate Cornish–Fisher Expansion
15.1 Introduction
15.2 Univariate Cornish–Fisher Expansion
15.3 Multivariate Cornish–Fisher Expansion
15.4 Application
15.5 Validity of Cornish–Fisher Expansion
15.6 Cornish–Fisher Expansion of Discrete Variables
References
Index