## Asymptotic StatisticsThis book is an introduction to the field of asymptotic statistics. The treatment is both practical and mathematically rigorous. In addition to most of the standard topics of an asymptotics course, including likelihood inference, M-estimation, the theory of asymptotic efficiency, U-statistics, and rank procedures, the book also presents recent research topics such as semiparametric models, the bootstrap, and empirical processes and their applications. The topics are organized from the central idea of approximation by limit experiments, which gives the book one of its unifying themes. This entails mainly the local approximation of the classical i.i.d. set up with smooth parameters by location experiments involving a single, normally distributed observation. Thus, even the standard subjects of asymptotic statistics are presented in a novel way. Suitable as a graduate or Master's level statistics text, this book will also give researchers an overview of research in asymptotic statistics. |

### From inside the book

Results 1-5 of 93

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**Theorem**8.6. Almost-Everywhere Convolution**Theorem***8.7. Local Asymptotic Minimax**Theorem***8.8. Shrinkage Estimators *8.9. Achieving the Bound *8.10. Large Deviations Problems 9. Limits of Experiments 9.1. Introduction 9.2. Asymptotic ... Page

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**theorem**, every uniformly tight sequence contains a weakly converging subsequence. Prohorov's**theorem**generalizes the Heine-Borel**theorem**from deterministic sequences Xn to random vectors. 2.4**Theorem**(Prohorov's**theorem**). Let Xn be ... Page

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**theorem**for convergence in probability By Again by the continuous-mapping**theorem**, Sn the central limit**theorem**) distribution. Finally, Slutsky's lemma gives that the sequence of t-statistics converges in distribution to N(0, var Y1 )/ ... Page

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**theorem**the converse is also true: Pointwise convergence of characteristic functions is equivalent to weak convergence. EeitTX 2.13**Theorem**(Lévy's continuity**theorem**). Let X n and X be random vectorsTX n converges in pointwise to a ... Page

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**theorem**, yields simple proofs of both the law of large numbers and the central limit**theorem**. 2.16 Proposition (Weak law of large numbers). Let Y1 ,..., Yn be i.i.d. random variables with characteristic function . Then is differentiable ...### Contents

Contiguity | |

Local Asymptotic Normality | |

Efficiency of Estimators | |

Limits of Experiments | |

Efficiency of Tests | |

Likelihood Ratio Tests | |

ChiSquare Tests | |

Problems | |

Empirical Processes | |

Functional Delta Method | |

Quantiles and Order Statistics | |

Bootstrap | |

Bayes Procedures | |

Projections | |

UStatistics | |

Rank Sign and Permutation Statistics | |

Relative Efficiency of Tests | |

Nonparametric Density Estimation | |

Semiparametric Models | |

References | |

Index | |

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### Common terms and phrases

alternative apply approximation argument assume assumption asymptotically normal bounded Chapter closed consider consistent constant contained continuous converges converges in distribution corresponding defined definition density depends derivative difference differentiable display distribution function efficient empirical equal equation equivalent Example exists expectation experiment finite fixed follows given gives hence independent inequality influence function instance integral interval known lemma likelihood ratio limit limit distribution linear matrix maximal maximum likelihood estimator mean measurable method null hypothesis observations obtain optimal parameter possible preceding probability problem projection proof prove quantile random range rank relative respect sample satisfies score function sequence Show side space square standard statistic subset sufficiently Suppose takes term theorem true uniform uniformly values variables variance vector yields zero