On the construction of least favourable pairs of distributions
On the construction of least favourable pairs of distributions
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关于最不利分布对的构造
DOI：
10.1007/bf00535275
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发表时间：
1978
期刊：
影响因子：
--
通讯作者：
F. Österreicher
中科院分区：
文献类型：
--
作者：
F. Österreicher
摘要Witting和Krafft在[4]中使用了凸编程技术，以将复合假设的测试问题简化为简单假设的测试问题。这是通过最不有利的分布对来实现的，这代表了适当方案的对偶的解。然而，在没有对假设进行进一步假设的情况下，结果以这种方式得出(参见Baumann[1]、Österreicher[6]和Kusolitsch andÖsterreicher[5])，实际影响较小。这是因为在这种情况下，最不有利的配对取决于测试问题的水平。Huber和Strassen在[3]中给出了避免这种情况的条件。这些条件利用了Choket意义下的2-交错容量。本文给出了当两个假设之一很简单时，构造最不有利分布的一个相当一般的原则。在单调似然比的情况下，该方法同样适用于局部变化模型和Prohorov邻域模型。对于“简单”情况--包括粗差模型和总变分模型，Huber在文[2]中给出了它的解--通过连续两次使用上述构造技术，得到了最不有利的一对。
SummaryConvex programming techniques were used by Witting and Krafft in [4] in order to reduce a testing problem for composite hypotheses to one for simple hypotheses. This is realized in terms of least favourable pairs of distributions, which represent the solution of the dual of a suitable program. Without further assumptions on the hypotheses, however, the results, derived that way (cf. Baumann [1], Österreicher [6] and Kusolitsch and Österreicher [5]), are of less practical impact. This is due to the fact that in this case the least favourable pairs depend on the level of the testing problem. Conditions avoiding this, were given by Huber and Strassen in [3]. These conditions make use of 2-alternating capacities in the sense of Choquet. The present paper offers a rather general principle of constructing the least favourable distribution in the case, when one of the two hypotheses is simple. This method works also for the local variation model and the Prohorov neighbourhood model in the case of monotone likelyhood ratio. For “simple” cases—subsuming the gross error model and the total variation model, for which the solution was given by Huber in [2]—a least favourable pair is obtained by using the mentioned technique of construction two times successively.