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单渐近最优检验：单因素分类的实用方法
- 研究问题
- 在多个正态总体可能同时存在均值和方差差异时，如何联合检验总体相等性，而不只在方差相同的前提下比较均值？
- 方法与思路
- 作者将均值相等检验与方差相等检验这两个独立检验通过 Fisher 方法组合，并借助相关最优性结果讨论 Bahadur 效率。
- 材料陈述的贡献
- 摘要称，组合检验简单、可使用现有统计表，且具有 Bahadur 意义下的渐近最优性；所给文字注明证明细节需向作者索取。
- 阅读关注点
- 核对两个检验独立性成立的条件、组合统计量与最优性适用范围，并特别检查重复测量等应用是否满足所需假设。
- 阅读边界
- 材料仅为版本未注明的 OpenAlex 索引摘要；最优性证明细节未提供，不能据此认定有限样本功效更高或推断刊本变化。
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.