A Simple and Asymptotically Optimal Test for the Equality of Normal Populations: A Pragmatic Approach to One-Way Classification
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正态总体相等性的简便渐近最优检验：单因素分类的实用方法
- 研究问题
- 当不同处理可能同时改变正态总体的均值和方差时，如何联合检验多个总体相等，补足只检验均值的单因素分析？
- 方法与思路
- 作者用 Fisher 方法合并两个独立检验：共同未知方差下的均值相等检验，以及均值可不同且未知时的方差相等检验。
- 材料陈述的贡献
- 摘要称组合检验具有 Bahadur 意义下的渐近最优性，并可借助现有统计表实施；最优性证明细节未在摘要中给出。
- 阅读关注点
- 核对两项检验独立性的适用条件、组合检验的显著性校准，以及 Bahadur 最优性论证是否完整且可供查验。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；无法验证最优性证明、有限样本表现或重复测量应用所需条件。
原文摘要
Snedecor and Cochran (1967, p. 324) observed that an application of different treatments to otherwise homogeneous experimental units often results in groups that are different not only in means but also in variances. The usual one-way classification procedure assumes a priori the homogeneity of variances among different groups and tests the equality of means only. Thus motivated by the problem of simultaneously testing the equality of means and the equality of variances of several normal populations, I suggest in this article a simple and an asymptotically optimal test. The suggested test is based on the combination of two independent tests, namely (a) the test for the equality of means given that all variances are equal but unspecified and (b) the test for the equality of variances when all means, not necessarily all equal, are unspecified. Tests (a) and (b) are combined by the Fisher method (1950, p. 99). Littell and Folks (1973) showed that the Fisher method of combining two or more independent tests is at least as good as any other optimal method. Using this fact and the results of Hsieh (1979) on the Bahadur optimality of the individual tests, the suggested test was found to be asymptotically optimal (details can be obtained from the author on request) in the sense of Bahadur (1960) efficiency. Further, the test can be applied easily using existing statistical tables. Examples where this test can be applied include the analysis of repeated measurements, the profile analysis of several groups, and, in general, the analysis of one-way classified experiments.