Asymptotic distribution of P values in composite null models
Asymptotic distribution of P values in composite null models
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DOI：
10.2307/2669750
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发表时间：
2000-12-01
影响因子：
3.7
通讯作者：
Ventura, V
中科院分区：
文献类型：
--
作者：
Robins, JM;van der Vaart, A;Ventura, V
我们通过计算p值来研究零模型H-0与数据的相容性;也就是说，在H-0下，给定的静止统计量T超过其观测值的概率。当零模型由单一分布组成时，容易获得p值，并且它在H-0下具有均匀分布。另一方面，当空模型依赖于未知的讨厌参数θ时，必须以某种方式设置摆脱θ，（例如，通过估计它）来计算a;o值。已经提出了各种建议来“移除”theta，每一个都产生不同的候选p值。但与简单情况不同的是，这些p值在空模型下通常不是均匀分布的。本文研究了它们在H-0下的渐近分布。我们证明了当检验统计量T的渐近均值依赖于θ时，Guttman和Rubin的后验预测p值和插件p值是保守的（即，它们的渐近分布比均匀分布更集中在1/2附近），后验预测p值更保守。相反。Bayarri和Berger的部分后验预测和条件预测p值是渐近一致的。此外，我们证明了Meng和Gelman及其同事的差异p值可以是保守的，即使在零模型下差异测度的均值为0。我们还描述了如何修改保守的p值，使其分布渐近均匀。
We investigate the compatibility of a null model H-0 with the data by calculating a p value; that is, the probability, under H-0, that a given rest statistic T exceeds its observed value. When the null model consists of a single distribution, the p value is readily obtained, and it has a uniform distribution under H-0. On the other hand, when the null model depends on an unknown nuisance parameter theta, one must somehow Set rid of theta, (e.g., by estimating it) to calculate a;o value. Various proposals have been suggested to "remove" theta, each yielding a different candidate p value. But unlike the simple case, these p values typically are not uniformly distributed under the null model. In this article we investigate their asymptotic distribution under H-0. We show that when the asymptotic mean of the test statistic T depends on theta, the posterior predictive p value of Guttman and Rubin, and the plug-in p value are conservative (i.e., their asymptotic distributions are more concentrated around 1/2 than a uniform), with the posterior predictive p value being the more conservative. In contrast. the partial posterior predictive and conditional predictive p values of Bayarri and Berger are asymptotically uniform. Furthermore, we show that the discrepancy p value of Meng and Gelman and colleagues can be conservative, even when the discrepancy measure has mean 0 under the null model. We also describe ways to modify the conservative p values to make their distributions asymptotically uniform.