Mathematical StatisticsCRC Press, 1999. jan. 27. - 592 oldal 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. |
Tartalomjegyzék
Caway hartsica | 1 |
tembulli fistribution | 37 |
Estimation of unknown parameters | 40 |
Pont eximation The main method of obtaining estimators Donsistency | 51 |
4 Consistency of M estimators | 60 |
15 The minimumdistance method | 65 |
Asymptotic properties of maximumlikelihood estimators Consistency | 74 |
En sommparing estimators | 80 |
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 |
Precise sample distributions and confidence intervals for normal populations | 236 |
normal distribution | 237 |
18 | 88 |
Asymptotic approach Asymptotic efficiency in the classes | 94 |
Klultidimensional ense | 100 |
Bayesian and minimax approaches to parameter estimation | 109 |
Sufficient statistics | 116 |
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 Refficient estimators | 139 |
Refficient and asymptotically Refficient estimators | 144 |
The Rao Cramer 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 |
Progettive id conditional expectations | 160 |
Pitman estimators are minimax | 162 |
On optimal estimators for the scale parameter | 163 |
29 General problem of equivariant estimation | 165 |
Integral Rao Cramer type inequality Criteria for estimators to be asymptotically Bayesian and minimax | 168 |
Main inequalities | 169 |
Inequalities for the case when the function qI 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 |
Estimates for the distribution and for the moments of a maximumlikelihood estimator Consistency of a maximumlikelihood estimator | 191 |
Asymptotic normality | 192 |
Testing two simple hypotheses | 249 |
Testing composite hypotheses Classes of optimal tests | 264 |
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 |
Testing the hypothesis that a sample belongs to a parametric family | 351 |
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 |
Sufficient statistics Efficient estimators Exponential families | 440 |
Rao Cramer inequality | 451 |
Gametheoretic approach to problems of mathematical statistics | 461 |
Bayes principle Complete class of decision functions | 476 |
Sufficiency unbiasedness invariance | 482 |
Asymptotically optimal estimators for an arbitrary loss function | 487 |
Optimal statistical tests for an arbitrary loss function The likelihood ratio | 497 |
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 |
| 568 | |
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a₁ arbitrary assertion assume asymptotically Bayesian asymptotically efficient asymptotically equivalent asymptotically minimax asymptotically normal asymptotically optimal Bayes test conditional expectation conditions RR confidence interval confidence set consider construct continuous convergence Corollary corresponding defined Definition denote density differentiable distribution function distribution Q efficient estimator equality equation equivariant estimators Example exists exponential family fe(x fo(x follows H₂ hypothesis H₁ independent integral invariant Lemma likelihood function likelihood ratio test lim sup M-estimators matrix maximum-likelihood estimator means measure minimal sufficient statistic minimax test multidimensional n₁ normal distribution observations obtain P₁ parameter powerful test priori distribution problem of testing proof of Theorem properties prove R-efficient random variables relation respect right-hand side sample satisfied second-order moments Section Subsection sufficient statistic Suppose testing H₁ testing the hypothesis uniformly most powerful values vector