Mathematical Statistics

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CRC Press, 1999. jan. 27. - 592 oldal
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A wide-ranging, extensive overview of modern mathematical statistics, this work reflects the current state of the field while being succinct and easy to grasp. The mathematical presentation is coherent and rigorous throughout.
The author presents classical results and methods that form the basis of modern statistics, and examines the foundations of estimation theory, hypothesis testing theory and statistical game theory. He then considers statistical problems for two or more samples, and those in which observations are taken from different distributions. Methods of finding optimal and asymptotically optimal statistical procedures are given, along with treatments of homogeneity testing, regression, variance analysis and pattern recognition. The author also posits a number of methodological improvements that simplify proofs, and brings together a number of new results which have never before been published in a single monograph.
 

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Tartalomjegyzék

Estimation of unknown parameters
40
Point estimation The main method of obtaining estimators Consistency
51
The maximumlikelihood method Optimality of maximumlikelihood
67
On comparing estimators
80
Comparing estimators in the parametric case Efficient estimators
91
Conditional expectations
101
Bayesian and minimax approaches to parameter estimation
109
Sufficient statistics
116
Estimates for the distribution and for the moments of a maximumlikelihood estimator Consistency of a maximumlikelihood estimator
191
Asymptotic normality
192
Asymptotic efficiency
194
Maximumlikelihood estimators are asymptotically Bayesian
195
Asymptotic properties of the likelihood ratio Further optimality properties of maximumlikelihood estimators
196
35 Approximate computation of maximumlikelihood estimators
204
The results of Sections 3335 for the multidimensional case
211
38 On statistical problems related to samples of random size Sequential estimation
226

23 Minimal sufficient statistics
122
Constructing efficient estimators via sufficient statistics Complete statistics
128
Multidimensional case
129
Complete statistics and efficient estimators
130
Exponential family
133
The RaoCramer inequality and efficient estimators
139
?efficient and asymptotically efficient estimators
144
The RaoCramer inequality in the multidimensional case
147
Some concluding remarks
151
27 Properties of the Fisher information
152
Multidimensional case
155
Fisher matrix and parameter change
157
28 Estimators of the shift and scale parameters Efficient equivariant estimators
158
Efficient estimator for the shift parameter in the class of equivariant estimators
159
Pitman estimators are minimax
162
On optimal estimators for the scale parameter
163
29 General problem of equivariant estimation
165
Integral RaoCramer type inequality Criteria for estimators to be asymptotically Bayesian and minimax
168
Main inequalities
169
Inequalities for the case when the function q8I8 is not differentiable
173
Some corollaries Criteria for estimators to be asymptotically Bayesian or minimax
174
Multidimensional case
177
Connection between the Hellinger and other distances and the Fisher information
180
Existence of uniform bounds for rAA2
181
Multidimensional case
182
5 Connection between the distances in question and estimators
183
32 Difference inequality of RaoCramer type
184
Auxiliary inequalities for the likelihood ratio Asymptotic properties of maximumlikelihood estimators
188
Main inequalities
189
Precise sample distributions and confidence intervals for normal populations
236
Uniformly most powerful tests
268
46 Unbiased tests
277
48 Connection with confidence sets
285
The Bayesian and minimax approaches to testing composite hypotheses
293
Likelihood ratio test
304
Testing composite hypotheses in the general case
315
Asymptotically optimal tests Likelihood ratio test as an asymptotically
323
Asymptotically optimal tests for testing close composite hypotheses
329
Asymptotic optimality properties of the likelihood ratio test which
336
The x2 test Testing hypotheses on grouped data
345
Robustness of statistical decisions
357
Statistical problems for two or more samples
368
Regression problems
390
Nonidentically distributed observations
411
Maximumlikelihood estimators The main principles of estimator comparison
432
Gametheoretic approach to problems of mathematical statistics
461
Theorems of GlivenkoCantelli type
505
Properties of conditional expectations
514
The law of large numbers and the central limit theorem Uniform versions
517
Some assertions concerning integrals depending on parameters
527
Inequalities for the distribution of the likelihood ratio in the multidimensional case
533
Proofs of two fundamental theorems of the theory of statistical games
537
Tables
543
Bibliographic comments
552
References
560
Notation
564
Index
568
Copyright

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