## Theory of StatisticsThe aim of this graduate textbook is to provide a comprehensive advanced course in the theory of statistics covering those topics in estimation, testing, and large sample theory which a graduate student might typically need to learn as preparation for work on a Ph.D. An important strength of this book is that it provides a mathematically rigorous and even-handed account of both Classical and Bayesian inference in order to give readers a broad perspective. For example, the "uniformly most powerful" approach to testing is contrasted with available decision-theoretic approaches. |

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### Contents

1 | |

Sufficient Statistics | 82 |

Decision Theory | 144 |

Hypothesis Testing | 214 |

Estimation | 296 |

Equivariance | 344 |

Large Sample Theory | 394 |

Hierarchical Models | 476 |

Sequential Analysis | 536 |

Measure and Integration Theory | 570 |

Probability Theory | 606 |

Mathematical Theorems Not Proven Here | 665 |

675 | |

689 | |

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### Common terms and phrases

a-field ancillary assume asymptotic Axiom Bayes factor Bayes rule Bayesian bootstrap Borel space calculate conditional distribution conditionally independent conditions of Theorem confidence set Consider Continuation of Example converges coordinates Corollary countable decision rule define Definition denote distribution given equal equivariant estimator exists exponential family finite follows formal Bayes rule function g Hence horse lotteries hypothesis improper prior integral interval invariant joint distribution Lebesgue measure Lemma level a test limn-co loss function matrix measurable function measure space nonnegative normal observations P-value parameter space parametric family posterior distribution posterior mean posterior probability posterior risk power function Pr(X prior distribution probability measure problem proof of Theorem random quantities random variables regular conditional reject H respect to Lebesgue risk function sample says Section sequence subset sufficient statistic Suppose that X1 test of H UMP level UMPU unbiased values variance vector