Testing Statistical Hypotheses
Testing Statistical Hypotheses
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DOI：
10.1007/978-1-4757-1923-9
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发表时间：
1959
影响因子：
7.3
通讯作者：
E. L. Lehmann
中科院分区：
文献类型：
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作者：
E. L. Lehmann
第3、4、5、6和7章涉及UMP、UMP无偏检验和UMP不变检验的推导。不幸的是，这种测试的存在被证明是限制基本上是一个参数的家庭单调似然比，指数家庭，和组家庭，分别。在适当的备选方案类别上，最大化最小或平均功效的检验相当普遍，但很难明确确定，第8章中的推导主要局限于不变性考虑适用的情况。尽管有其局限性，这些方法已证明其价值的应用到大类的重要情况。另一方面，它们不太可能适用于复杂的新问题。对于这种情况，需要的是一种更简单、不那么详细、更普遍适用的提法。制定和实施这一办法将是其余各章的主题。它取代了渐近最优性的最优性，通过嵌入的实际情况下，增加样本量的情况序列，并应用最优性的极限情况。这些极限往往是一种简单的类型，在前面的章节中已经建立了最优性。渐近最优性的一个特征是它不是指一个单一的检验，而是指一系列的检验，尽管这种区别经常被抑制。一个重要的后果是，渐近最优的程序，不像大多数最优的程序，在小样本的方法，是不是唯一的，因为许多不同的序列有相同的限制。事实上，完全不同的构造方法可能导致渐近最优的过程。下面是一些具体的例子，需要记住的是，在有限样本的考虑无法提供最佳程序，但大样本的方法将更成功。
Chapters 3, 4, 5, 6, and 7 were concerned with the derivation of UMP, UMP unbiased, and UMP invariant tests. Unfortunately, the existence of such tests turned out to be restricted essentially to one-parameter families with monotone likelihood ratio, exponential families, and group families, respectively. Tests maximizing the minimum or average power over suitable classes of alternatives exist fairly generally, but are difficult to determine explicitly, and their derivation in Chapter 8 was confined primarily to situations in which invariance considerations apply. Despite their limitations, these approaches have proved their value by application to large classes of important situations. On the other hand, they are unlikely to be applicable to complex new problems. What is needed for such cases is a simpler, less detailed, more generally applicable formulation. The development and implementation of such an approach will be the subject of the remaining chapters. It replaces optimality by asymptotic optimality obtained by embedding the actual situation in a sequence of situations of increasing sample size, and applying optimality to the limit situation. These limits tend to be of a simple type for which optimality has been established in earlier chapters.A feature of asymptotic optimality is that it refers not to a single test but to a sequence of tests, although this distinction will often be suppressed. An important consequence is that asymptotically optimal procedures—unlike most optimal procedures in the small-sample approach—are not unique since many different sequences have the same limit. In fact, quite different methods of construction may lead to procedures which are asymptotically optimal. The following are some specific examples to keep in mind where finite-sample considerations fail to provide optimal procedures, but for which a large-sample approach will be more successful.