The Asymptotic Powers of Certain Tests Based on Multiple Correlations
基于复相关系数的若干检验的渐近功效AI 解读 ↓
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基于复相关系数的若干检验的渐近功效
- 研究问题
- 如何比较基于复相关系数的检验的渐近功效，尤其在两个检验使用不同数量的回归变量、自由度不一致时？
- 方法与思路
- 作者扩展相同极限分布形式下的相对效率比较；自由度不同时使用非中心卡方分布，并用原假设下协方差矩阵表达相关距离。
- 材料陈述的贡献
- 摘要称，同自由度时可推广 Pitman 比较准则；自由度不同时该准则不能直接使用，但部分情形仍能获得明确比较结果。
- 阅读关注点
- 应核对局部备择与渐近功效的定义、复相关统计量的卡方近似，以及自由度变化时距离参数和协方差矩阵的作用。
- 阅读边界
- 仅依据所给 Crossref 摘要版本；具体比较结论与条件未提供，不能将相同自由度下的效率准则直接用于不同自由度。
原文摘要
Summary Pitman (1948) has proposed a criterion of the relative power of two tests of an hypothesis specified by a single parameter, based on the estimating efficiencies of the two test statistics, in case these test statistics are asymptotically normal. This criterion can be immediately generalized to many cases where the two statistics have limiting distributions of the same analytical form and, in particular, enables two tests based on statistics which are asymptotically distributed as chi-square to be compared, provided the numbers of degrees of freedom are the same. This was already recognized by Andrews (1954) who showed that the efficiencies of certain rank tests for analysis of variance could be compared in this way. Bradley (1955) also used the concept of asymptotic relative efficiency in relation to rank tests based on statistics which are asymptotically distributed as chi-square. The first section of this paper is therefore merely a slight generalization of Andrew’s and Bradley’s work. The asymptotic powers of multiple correlations may then be compared since these statistics are asymptotically equivalent to statistics distributed as chi-square. In some cases which have arisen the two multiple correlations to be compared have not been based on the same numbers of regressors. In such cases the comparison requires the use of tables of non central chi-square and Pitman’s criterion is no longer directly applicable. However the distances, on which the asymptotic powers of the tests depend, can be simply expressed in terms of the covariance matrix of the regressors on the null hypothesis and in some instances clear-cut results may still be simply derived. The applications in this paper relate to previous work by the author (Hannan (1955a), (1955b), (1956)).