## Testing Statistical HypothesesThe Third Edition of Testing Statistical Hypotheses brings it into consonance with the Second Edition of its companion volume on point estimation (Lehmann and Casella, 1998) to which we shall refer as TPE2. We won’t here comment on the long history of the book which is recounted in Lehmann (1997) but shall use this Preface to indicate the principal changes from the 2nd Edition. The present volume is divided into two parts. Part I (Chapters 1–10) treats small-sample theory, while Part II (Chapters 11–15) treats large-sample theory. The preface to the 2nd Edition stated that “the most important omission is an adequate treatment of optimality paralleling that given for estimation in TPE.” We shall here remedy this failure by treating the di?cult topic of asymptotic optimality (in Chapter 13) together with the large-sample tools needed for this purpose (in Chapters 11 and 12). Having developed these tools, we use them in Chapter 14 to give a much fuller treatment of tests of goodness of ?t than was possible in the 2nd Edition, and in Chapter 15 to provide an introduction to the bootstrap and related techniques. Various large-sample considerations that in the Second Edition were discussed in earlier chapters now have been moved to Chapter 11. |

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### Contents

3 | |

28 | |

Uniformly Most Powerful Tests | 56 |

Theory and First Applications | 110 |

Applications to Normal Distributions | 150 |

tions | 157 |

Invariance | 212 |

Linear Hypotheses | 277 |

Conditional Inference | 392 |

LargeSample Theory | 417 |

Quadratic Mean Differentiable Families | 482 |

Large Sample Optimality | 527 |

Testing Goodness of Fit | 583 |

General Large Sample Methods | 631 |

A Auxiliary Results | 692 |

References | 702 |

The Minimax Principle | 319 |

Multiple Testing and Simultaneous Inference | 348 |

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### Common terms and phrases

alternatives approximation Assume assumptions binomial bootstrap conditional distribution confidence intervals confidence sets continuous convergence covariance matrix cumulative distribution function defined degrees of freedom denote depends determined distribution function equivalent equivariant estimator Example exists a UMP exponential family finite fixed follows given hence hypothesis H implies independently distributed Lebesgue measure Lemma Let X1 likelihood ratio limiting power linear matrix maximal invariant mean measure normal distribution null hypothesis observations obtained optimal parameter permutation permutation test probability density problem of testing procedure proof random variables real-valued rejection region respect result risk function sample satisfying Section space subset sufficient statistic Suppose X1 t-test test of H test statistic testing H Theorem transformations UMP invariant test UMP test UMP unbiased test variance vector Wald test zero