## 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. |

### Contents

Stochastic Convergence | 5 |

Delta Method | 25 |

Moment Estimators | 35 |

Contiguity | 85 |

Local Asymptotic Normality | 92 |

Efficiency of Estimators | 108 |

Limits of Experiments | 125 |

Bayes Procedures | 138 |

Likelihood Ratio Tests | 227 |

ChiSquare Tests | 242 |

Stochastic Convergence in Metric Spaces | 255 |

Empirical Processes | 265 |

Functional Delta Method | 291 |

Quantiles and Order Statistics | 304 |

Bootstrap | 326 |

Nonparametric Density Estimation | 341 |

Projections | 153 |

UStatistics | 161 |

Rank Sign and Permutation Statistics | 173 |

Relative Efficiency of Tests | 192 |

Efficiency of Tests | 215 |

Semiparametric Models | 358 |

433 | |

439 | |

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### Common terms and phrases

approximation asymptotic variance asymptotically efficient asymptotically normal asymptotically optimal bootstrap bounded central limit theorem Chapter chi-square consistent continuous converges in distribution converges in probability converges to zero covariance matrix defined delta method density derivative differentiable in quadratic distribution function empirical distribution empirical process equal equation estimator sequence Example exists finite Fisher information fixed follows function F given H₁ hence inequality kernel large numbers Let X1 likelihood ratio statistic limit distribution limit experiment linear M-estimators maximal maximum likelihood estimator mean zero measurable function norm normal distribution null hypothesis observations orthogonal power function preceding display proof quadratic mean quantile random sample random variables random vectors rank statistics sample X1 satisfies score function sequence of tests Show space standard normal submodel subset Suppose tangent set U-statistic underlying distribution uniform uniformly values Y₁