Contributions on Theory of Mathematical Statistics

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