## Mathematical StatisticsA 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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### Contents

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 |

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 |

Precise sample distributions and confidence intervals for normal populations | 236 |

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 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 qOIO 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 |

normal distribution | 237 |

Uniformly most powerful tests | 268 |

construction of most powerful tests and uniformly most powerful tests | 275 |

47 Invariant tests | 281 |

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 |

Statistical problems for two or more samples | 368 |

Regression problems | 390 |

Nonidentically distributed observations | 411 |

Sufficient statistics Efficient estimators Exponential families | 440 |

RaoCramer inequality | 451 |

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 |

63 | 552 |

560 | |

67 | 562 |

Notation | 564 |

568 | |

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### Common terms and phrases

addition approach approximation arbitrary assertion assume assumption asymptotically asymptotically normal Bayesian belongs bounded called Chapter close complete confidence interval consider consistent construct continuous convergence Corollary corresponding decision defined Definition Denote density depend determined differentiable distribution efficient equal equation equivalent Example exists fact fixed fo(X follows function given holds hypothesis implies independent inequality instance integral introduced invariant Lemma likelihood ratio limit loss M-estimators matrix maximum-likelihood estimator means measure method minimal minimax moments natural Note observations obtain optimal parameter possible powerful test presented probability problem Proof properties prove random variables relation remains REMARK respect sample satisfied Section side similar solution strategy Subsection sufficient statistic Suppose testing the hypothesis Theorem transformation unbiased uniform unknown valid values vector