Some Scheffé-Type Tests for Some Behrens-Fisher-Type Regression Problems
若干 Behrens–Fisher 型回归问题的 Scheffé 型检验AI 解读 ↓
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若干 Behrens–Fisher 型回归问题的 Scheffé 型检验
- 研究问题
- 当两组正态回归误差的方差不同时，如何检验两条回归线是否平行，以及已知平行时是否完全重合？
- 方法与思路
- 作者推广 Scheffé 的思路，以通常的 t 统计量构造随机化和非随机化检验，并考虑 Welch–Hájek 方法。
- 材料陈述的贡献
- 摘要称，文章处理简单与多元回归，提供检验和置信界，并对已知平行情形下两线距离给出极小极大估计。
- 阅读关注点
- 应核对不同检验的构造、误差假设与水平控制，以及置信界和极小极大估计所针对的参数及损失函数。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；检验公式和比较性能未提供，教育应用举例不构成因果识别证据。
原文摘要
In educational and psychological applications as well as in other applications, it may be necessary to make certain comparisons of two regression lines when the variances are unequal. Such problems arise, for example, in studies comparing two alternative curriculums or two different teaching methods. By generalizing an idea which Scheffé used to obtain a test for the Behrens-Fisher problem, this paper develops some tests for comparing two regression lines when the two sets of error terms are normally distributed but with two different variances. Scheffé's test itself is a randomized test, but in this paper we present both randomized and non-randomized tests. Both simple and multiple regression are considered, but the simple regression tests are computationally easier than the multiple regression tests. The basic test statistic which is used is the ordinary t-statistic. Essentially two types of problems are dealt with: (A) determining whether the two regression lines are identical when they are known to be parallel; and (B) determining whether the two regression lines are parallel. Confidence bounds as well as tests of hypotheses are available. For Problem A, a minimax estimator of the distance between the two lines is obtained. In addition to the Scheffé-type tests, we also consider some tests based on an approach of Welch and Hájek. A numerical example is presented.