## Asymptotic Theory of Testing Statistical Hypotheses: Efficient Statistics, Optimality, Power Loss, and DeficiencyThe series is devoted to the publication of high-level monographs and surveys which cover the whole spectrum of probability and statistics. The books of the series are addressed to both experts and advanced students. |

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### Contents

Asymptotic expansions under alternatives | 49 |

Power loss | 71 |

Structure and Limit | 74 |

Edgeworth expansion for the likelihood ratio | 125 |

A LeCams Third Lemma | 149 |

B Convergence rate under alternatives | 157 |

Proof of Theorem 1 3 1 | 179 |

E Edgeworth expansions | 185 |

F Proof of Lemmas 2 6 12 6 5 | 201 |

G Proof of Lemmas 3 7 13 7 5 | 215 |

H Asymptotically complete classes | 229 |

Higher order asymptotics for R L and statistics | 239 |

255 | |

262 | |

271 | |

### Common terms and phrases

absolutely continuous Albers alternatives An(t Assume Assumption asymptotic expansions asymptotic relative efficiency asymptotically efficient test asymptotically normal Bening Berry-Esseen bound Bickel Borel sets characteristic function Chibisov combinations of order complete class consider convergence Cramer deficiency defined denote density p(x derivative distributed random variables distribution function Edgeworth expansion empirical distribution function exists Fn(x follows formula Hajek and Sidak Helmers Hence hypothesis identically distributed random implies independent identically distributed inequality integral L-statistics Let Conditions Let us prove likelihood ratio linear rank statistic Neyman-Pearson Lemma obtain order statistics parameter Pfanzagl polynomials power function probability measures problem Proof of Lemma proof of Theorem Proposition 4.1.1 pseudo-cumulants r-system rank statistics regularity conditions remainder term respect sample satisfied Section sequence stochastic expansion sums of independent terms of order test based uniformly uniformly integrable van Zwet variance vector yields Zwet

### Popular passages

Page 255 - A note on the Edgeworth expansion for the Kendall rank correlation coefficient. Ann.