An approximate degrees of freedom solution to the multivariate Behrens-Fisher problem*
An approximate degrees of freedom solution to the multivariate Behrens-Fisher problem*
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DOI：
10.1093/biomet/52.1-2.139
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发表时间：
1965-06
期刊：
影响因子：
2.7
通讯作者：
Ying Yao
中科院分区：
文献类型：
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作者：
Ying Yao
基于两个独立样本比较两个总体的平均值是统计学中最古老的问题之一。事实上，它一直是许多推理方法以及各种实际问题分析方法的试验场。单变量问题首先由 Behrens (1929) 研究，Fisher (1935) 根据基准理论提出了他的解决方案。韦尔奇在置信理论框架中对其进行了研究，并提供了“近似自由度”解以及渐近级数解（1936、1947）。许多其他人已经研究了这个主题，Jeifreys (1940)、Scheff6 (1943)、McCullough、Gurland & Rosenberg (1960)、Banerjee (1961) 和 Savage (1961) 也提出了各种方法。在 Behrens-Fisher 问题的多元扩展中，Bennett (1951) 扩展了 Scheff6 解，James (1954) 扩展了 Welch 级数解。本文研究了 Tukey (1959) 提供的 Welch“近似自由度”(APDF) 解的扩展，并讨论了该新 APDF 解的蒙特卡洛抽样研究结果及其与 James 级数解的比较。
The comparison of the means of two populations on the basis of two independent samples is one of the oldest problems in statistics. Indeed, it has been a testing ground for many methods of inference as well as for a variety of analytic approaches to practical problems. The univariate problem was first studied by Behrens (1929) and his solution was presented by Fisher (1935) in terms of the fiducial theory. Welch studied it in the confidence theory framework and provided an 'approximate degrees of freedom' solution as well as an asymptotic series solution (1936, 1947). Many others have investigated this topic and various methods of approach were also suggested by Jeifreys (1940), Scheff6 (1943), McCullough, Gurland & Rosenberg (1960), Banerjee (1961), and Savage (1961). In the multivariate extension of the Behrens-Fisher problem, Bennett (1951) has extended the Scheff6 solution, and James (1954) the Welch series solution. The present paper studies an extension of the Welch 'approximate degrees of freedom' (APDF) solution provided by Tukey (1959), and discusses the results of a Monte Carlo sampling study on this new APDF solution and its comparison with the James series solution.