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Robust and Non-Robust Models in Statistics
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Authors: Lev B. Klebanov (Charles University, Czech Republic) Svetlozar T. Rachev (University of Karlsruhe, Germany) Frank J. Fabozzi (Yale School of Management, New Haven, CT) 
Editors: Mathematics Research Developments
Book Description:
In this book the authors consider so-called ill-posed problems and stability in statistics. Ill-posed problems are certain results where arbitrary small changes in the assumptions lead to unpredictable large changes in the conclusions. In a companion problem published by Nova, the authors explain that ill-posed problems are not a mere curiosity in the field of contemporary probability. The same situation holds in statistics. The objective of the authors of this book is to (1)identify statistical problems of this type, (2) find their stable variant, and (3)propose alternative versions of numerous theorems in mathematical statistics.
The layout of the book is as follows. The authors begin by reviewing the central pre-limit theorem, providing a careful definition and characterization of the limiting distributions. Then, they consider pre-limiting behavior of extreme order statistics and the connection of this theory to survival analysis. A study of statistical applications of the pre-limit theorems follows. Based on these theorems, the authors develop a correct version of the theory of statistical estimation, and show its connection with the problem of the choice of an appropriate loss function. As It turns out,a loss function should not be chosen arbitrarily. As they explain, the availability of certain mathematical conveniences (including the correctness of the formulation of the problem estimation) leads to rigid restrictions on the choice of the loss function. The questions about the correctness of incorrectness of certain statistical problems may be resolved through appropriate choice of the loss function and/or metric on the space of random variables and their characteristics (including distribution functions, characteristic functions, and densities). Some auxiliary results from the theory of generalized functions are provided in an appendix.

Table of Contents:

About the authors

Part 1. Models in Statistical Estimation Theory

Chapter 1. Ill-posed problems;pp. 5-27

1. Introduction and motivating examples
2. Central Pre-Limit Theorem
3. Sums of a random number of random variables
4. Local pre-limit theorems and their applications to nance
5. Pre-limit theorem for extremums
6. Relations with robustness of statistical estimators
7. Statistical estimation for non-smooth densities
8. Key points of this chapter

Chapter 2. Loss functions and the restrictions imposed on the model;pp. 29-55

1. Introduction
2. Reducible families of functions
3. The classification of classes of estimators by their completeness types
4. An example of a loss function
5. Concluding remarks
6. Key points of this chapter

Chapter 3. Loss functions and the theory of unbiased estimation;pp. 57-93

1. Introduction
2. Unbiasedness, Lehmann's unbiasedness, and W1-unbiasedness
3. Characterizations of convex and strictly convex loss functions
4. Unbiased estimation, universal loss functions, and optimal subalgebras
5. Matrix-valued loss functions
6. Concluding remarks
7. Key points of this chapter

Chapter 4. Sufficient statistics;pp. 95-114

1. Introduction
2. Completeness and Sufficiency
3. Sufficiency when nuisance parameters are present
4. Bayes estimators independent of the loss function
5. Key points of this chapter

Chapter 5. Parametric inference;pp. 115-131

1. Introduction
2. Parametric Density Estimation versus Parameter Estimation
3. Unbiased parametric inference
4. Bayesian parametric inference
5. Parametric density estimation for location families
6. Key points of this chapter

Chapter 6. Trimmed, Bayes, and admissible estimators;pp. 133-138

1. Introduction
2. A trimmed estimator cannot be Bayesian
3. Linear regression model: Trimmed estimators and admissibility
4. Key points of this chapter

Chapter 7. Characterization of Distributions and Intensively Monotone Operators;pp. 139-169

1. Introduction
2. The uniqueness of solutions of operator equations
3. Examples of intensively monotone operators
4. Examples of strongly E-positive families
5. A generalization of Cramer's and Polya's theorems
6. Random linear forms
7. Some problems related to reliability theory
8. Key points of this chapter

Part 2. Robustness For a Fixed Number Of The Observations

Chapter 8. Robustness of Statistical Models;pp. 173-217

1. Introduction
2. Preliminaries
3. Robustness in statistical estimation and the loss function
4. A linear method of statistical estimation
5. Polynomial and modi fied polynomial Pitman estimators
6. Non-admissibility of polynomial estimators of location
7. The asymptotic e-admisibility of the polynomial Pitman's estimators of the location parameter
8. Key points of this chapter

Chapter 9. Entire function of fi nite exponential type and estimation of density function;pp. 219-234
1. Introduction
2. Main definitions
3. Fourier transform of the functions from M vp
4. Interpolation formula
5. Inequality of different metrics
6. Valle'e Poussin kernels
7. Key points of this chapter

Part 3. Metric Methods in Statistics

Chapter 10. N-Metrics in the Set of Probability Measures;pp. 237-253

1. Introduction
2. A Class of Positive Definite Kernels in the Set of Probabilities and N-distances
3. m-negative Definite Kernels and Metrics
4. Statistical Estimates obtained by the Minimal Distances Method
5. Key points of this chapter

Chapter 11. Some Statistical Tests Based on N-Distances;pp. 255-264
1. Introduction
2. Multivariate two-sample test
3. Test for two distributions to belong to the same additive type
4. Some Tests for Observations to be Gaussian
5. A Test for Closeness of Probability Distributions
6. Key points of this chapter

Appendix A. Generalized Functions;pp. 265-275

Appendix B. Positive and Negative De finite Kernels and Their Properties;pp. 277-293

Bibliography 237
Author Index 247

Index pp.315-317

   Binding: Hardcover
   Pub. Date: 2009
   Pages: 317.pp
   ISBN: 978-1-60741-768-2
   Status: AV
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
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