Estimation of Dependences Based on Empirical Data

Etukansi
Springer Science & Business Media, 28.9.2006 - 505 sivua
Twenty-?ve years have passed since the publication of the Russian version of the book Estimation of Dependencies Based on Empirical Data (EDBED for short). Twen- ?ve years is a long period of time. During these years many things have happened. Looking back, one can see how rapidly life and technology have changed, and how slow and dif?cult it is to change the theoretical foundation of the technology and its philosophy. I pursued two goals writing this Afterword: to update the technical results presented in EDBED (the easy goal) and to describe a general picture of how the new ideas developed over these years (a much more dif?cult goal). The picture which I would like to present is a very personal (and therefore very biased) account of the development of one particular branch of science, Empirical - ference Science. Such accounts usually are not included in the content of technical publications. I have followed this rule in all of my previous books. But this time I would like to violate it for the following reasons. First of all, for me EDBED is the important milestone in the development of empirical inference theory and I would like to explain why. S- ond, during these years, there were a lot of discussions between supporters of the new 1 paradigm (now it is called the VC theory ) and the old one (classical statistics).

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Sisältö

The Problem of Estimating Dependences from
1
The Problem of Interpreting Results of Indirect Experiments
8
The Accuracy of Estimating Dependences on the Basis
15
A2 Problems Well Posed in Tihonovs Sense
22
Uniform Proximity between Empirical Means and Mathematical Expectations
39
A Generalization of the GlivenkoCantelli Theorem and the Problem of Pattern Recognition
41
Remarks on Two Procedures for Minimizing Expected Risk on the Basis of Empirical Data
42
Methods of Parametric Statistics for the Pattern Recognition Problem
45
A Deterministic Statement of the Problem
144
Upper Bounds on Error Probabilities
146
An Enet of a Set
149
Necessary and Sufficient Conditions for Uniform Convergence of Frequencies to Probabilities
152
Properties of Growth Functions
154
Bounds on Deviations of Empirically Optimal Decision Rules
155
Remarks on the Bound on the Rate of Uniform Convergence of Frequencies to Probabilities
158
Remark on the General Theory of Uniform Estimating of Probabilities
159

The Pattern Recognition Problem 2 Discriminant Analysis
46
Decision Rules in Problems of Pattern Recognition
49
Evaluation of Qualities of Algorithms for Density Estimation
51
The Bayesian Algorithm for Density Estimation
52
Bayesian Estimators of Discrete Probability Distributions
54
Bayesian Estimators for the Gaussian Normal Density
56
Unbiased Estimators 9 Sufficient Statistics 10 Computing the Best Unbiased Estimator
66
The Problem of Estimating the Parameters of a Density
70
The MaximumLikelihood Method 13 Estimation of Parameters of the Probability Density Using the MaximumLikelihood Method
73
Methods of Parametric Statistics for
74
Problem of Regression Estimation 1 The Scheme for Interpreting the Results of Direct Experiments
81
A Remark on the Statement of the Problem of Interpreting the Results of Direct Experiments
83
Density Models
84
Extremal Properties of Gaussian and Laplace Distributions
87
On Robust Methods of Estimating Location Parameters
91
Robust Estimation of Regression Parameters
96
Robustness of Gaussian and Laplace Distributions
99
Classes of Densities Formed by a Mixture of Densities
101
Densities Concentrated on an Interval
103
Robust Methods for Regression Estimation 101 103
105
The Problem of Estimating Regression Parameters
112
The Theory of Normal Regression
113
Methods of Estimating the Normal Regression that are Uniformly Superior to the LeastSquares Method
115
A Theorem on Estimating the Mean Vector of a Multivariate Normal Distribution
120
The GaussMarkov Theorem
125
Best Linear Estimators
127
Criteria for the Quality of Estimators
128
Evaluation of the Best Linear Estimators
130
Utilizing Prior Information
134
A Method of Minimizing Empirical Risk for the Problem of Pattern Recognition
139
Uniform Convergence of Frequencies of Events to Their Probabilities
141
A Particular Case
142
Sufficient Conditions
162
A2 The Growth Function
163
A3 The Basic Lemma
168
A4 Derivation of Sufficient Conditions
170
A5 A Bound on the Quantity I
173
A6 A Bound on the Probability of Uniform Relative Deviation
176
A Method of Minimizing Empirical Risk for the Problem of Regression Estimation
181
A Particular Case
183
A Generalization to a Class with Infinitely Many Members
186
The Capacity of a Set of Arbitrary Functions
188
Uniform Boundedness of a Ratio of Moments
191
Two Theorems on Uniform Convergence
192
Theorem on Uniform Relative Deviation
195
Remarks on a General Theory of Risk Estimation
202
Appendix to Chapter 7 Theory of Uniform Convergence
206
A3 Eextension of a Set
215
A7 Corollaries
231
Solution of Illposed Problems Interpretation
267
Appendix to Chapter 9 Statistical Theory of Regularization
308
Estimation of Functional Values at Given Points
312
A Bound on the Uniform Relative Deviation of Means
318
Selection of a Sample for Estimating Values of
327
Estimation of Values of an Indicator Function in the Class
334
The Problem of Finding the Best Point of a Given Set
341
Appendix to Chapter 10 Taxonomy Problems
347
Algorithms for Estimating Nonindicator
370
Bibliographical Remarks
384
Bibliography
391
115
396
Index
397
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