Bahadur Efficiency of Tests of Separate Hypotheses and Adaptive Test Statistics
分离假设检验与自适应检验统计量的 Bahadur 效率AI 解读 ↓
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分离假设检验与自适应检验统计量的 Bahadur 效率
- 研究问题
- 在分离假设及未知干扰参数下，检验能否对不同干扰参数都保持渐近效率，经典 Cox 检验又可能怎样失效？
- 方法与思路
- 作者在指数族中以 Bahadur 渐近最优性分析平滑检验统计量，推导精确斜率表达及自适应检验存在条件。
- 材料陈述的贡献
- 摘要称，对数正态与指数分布的比较中 Cox 检验可能具有零 Bahadur 效率，并给出自适应统计量存在的充要条件。
- 阅读关注点
- 应核对分离假设的定义、斜率公式的正则条件，以及最不利备择、干扰参数与自适应最优性之间的关系。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；不能推断刊本变化。摘要未列出充要条件，零效率结论也不能推广至所有 Cox 检验问题。
原文摘要
The notion of separability of hypotheses was first introduced by Cox (1961); two families of hypotheses are separate if no distribution in one family can be obtained as a limit of distributions from the other family. For testing separate hypotheses, Cox (1961, 1962) suggested the test based on the comparison of the logarithm of the likelihood ratio with its expected value (or estimate thereof) under each hypothesis. The tests of such hypotheses are required to have high power against the specified alternatives; when nuisance parameters are present, this concept leads to the notion of an adaptive test, which by definition must be asymptotically efficient for any value of the unknown nuisance parameter. A testing problem of separate hypotheses for an exponential family is studied from the standpoint of Bahadur asymptotic optimality. A formula for the Bahadur-exact slope of any smooth test statistic is obtained, and in the example of testing lognormality versus exponentiality, it is shown that Cox's test can have zero Bahadur efficiency. A necessary and sufficient condition for the existence of an adaptive test statistic is derived using a result of Rubin and Rukhin (1983). This condition is met if the test is required to be asymptotically optimal at the least favorable alternative; that is, at the point that is the “closest” in terms of Kullback-Leibler divergence.