THE FREQUENCY DISTRIBUTION OF THE PRODUCT-MOMENT CORRELATION COEFFICIENT IN RANDOM SAMPLES OF ANY SIZE DRAWN FROM NON-NORMAL UNIVERSES
THE FREQUENCY DISTRIBUTION OF THE PRODUCT-MOMENT CORRELATION COEFFICIENT IN RANDOM SAMPLES OF ANY SIZE DRAWN FROM NON-NORMAL UNIVERSES
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DOI：
10.1093/biomet/38.1-2.219
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发表时间：
1951-01-01
期刊：
影响因子：
2.7
通讯作者：
GAYEN, AK
中科院分区：
文献类型：
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作者：
GAYEN, AK
Quensel(1938)从二元Gram-Charlier总体(另一种形式，下面的公式(18))导出了样本中r的频率密度，但正如他所说的，结果仅在父母相关p为零时适用。在完全独立变量的特殊情况下，他从他研究的公式中观察到了r的频率分布的性质。本文致力于非正态r分布的理论研究，在一般非零的情况下，即父母相关p存在且不一定小的情况下。考虑用二元A型曲面的Edgeworth形式表示的总体(包括A20、A30中的项)。032和A03)，得到了r的频率密度的数学形式。如果较高的半不变量除A40、A31、22、13和A04外，以及A20、A30、A21。A.总体的A12和A23可以忽略不计，那么导出的分布对任何大小的样本都适用。另一方面，如果样本相当大，则该公式对于任何形式的母总体都是渐近成立的。因此，对于中等大小的样本，所得到的分布具有相当大的适用范围并不是不可能的，只要所考虑的那些以外的更高的半不变量总是很小的。在这项工作中，假设总体的p和A的值是已知的。相关系数的Fisher对数变换即1+r和1+pz=S对数1-r1-p的应用可能性
Quensel (1938) derived the frequency density of r in samples from the bivariate Gram-Charlier population (alternative form, formula (18) below), but the result, as has been stated by him, is applicable only when the parental correlation p is zero. From the derived formulae of his study he made some observations as to the nature of the frequency distribution of r, in special cases of completely independent variates. The present paper is devoted to a theoretical study of the distribution of non-normal r, in the general non-null case, ie the case in which the parental correlation p exists and is not necessarily small. Considering the population to be expressed by the Edgeworth form of the bivariate Type A surface (including terms as far as those in A20, A30. 032 and A03) the mathematical form of the frequency density of r is obtained. If higher semi-invariants other than A40, A31 22 13 and A04, and A20, A30 A21. A. A12 and A23 of the population are negligible, then the derived distribution holds good for any size of sample. On the other hand, if the samples are fairly large, the formula will hold asymptotically for any form of parent population. Thus it is not unlikely that for samples of moderate size, the distribution obtained has quite an extended range of applicability, provided always the higher semi-invariants other than those considered are small. Throughout this work the values of p and of the A's of the population have been assumed to be known. The possibility of the application of Fisher's logarithmic transformation of the coefficients of correlation, namely 1+ r and 1+ p z= S loge 1-r 1-p