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).

Kirjan sisältä

Sisältö

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
The Problem of Estimating the Probability Density
36
Methods of Parametric Statistics for
45
Evaluation of Qualities of Algorithms for Density Estimation
51
Unbiased Estimators
63
The Problem of Estimating the Parameters of a Density
70
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 г
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

Estimation of Parameters of the Probability Density Using
76
A Remark on the Statement of the Problem of Interpreting
83
On Robust Methods of Estimating Location Parameters
91
Robustness of Gaussian and Laplace Distributions
99
Robust Methods for Regression Estimation
105
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
A Deterministic Statement of the Problem
144
Upper Bounds on Error Probabilities
146
An ɛnet 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
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 extension of a Set
214
A7 Corollaries
228
Solution of Illposed Problems Interpretation
267
Proofs of the Theorems
275
Methods of Polynomial and Piecewise Polynomial Approximations
285
The Problem of Probability Density Estimation
292
Density Estimation Using Parzens Method
301
Appendix to Chapter 9 Statistical Theory of Regularization
308
Estimation of Functional Values at Given Points
312
Appendix to Chapter 10 Taxonomy Problems
347
Algorithms for Estimating Nonindicator
370
Bibliographical Remarks
384
Bibliography
391
Index
397
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