从统计学史解析“自由度”概念

自由度是统计学中一个十分重要而又长期没有被圆满解释的概念。对此,从统计学史角度,对皮尔逊、费歇尔有关自由度问题争论原始文献细致考察,彻底澄清了自由度概念的内涵及与其相关的统计思想,弥补了Fienberg、Stigler与陈希孺已有解释的缺陷。研究表明：皮尔逊关于卡方检验中无论总体分布已知还是其来自于样本推断统计量都具有同一分布的错误判断,导致卡方检验的准确性出现偏差,这种偏差虽被同时代少数几个统计学家察觉但他们却无法解释其根源。费歇尔提出自由度概念并结合n维几何、假设检验与最大似然方法的论证不仅修正了皮尔逊的错误,也完善了从样本统计量估计总体参数的数理逻辑。

An Analysis of "Degrees of Freedom" from the History of Statistics

＂Degrees of freedom＂ is a concept of central importance to statistics though which has never been satisfactorily interpreted.Therefore,this paper thoroughly clarifies the connotation of degrees of freedom in which contains enormous statistical thinking and makes up for some defects in former researches by Fienberg,Stigler and Chen Xi-ru,through the approach of history of statistics,and the careful investigation based on the primary literature about the debates of degrees of freedom between Karl Pearson and R.A.Fisher.The results suggest that there was a error in Pearson＇s Chi-square Test by his misjudgment while the theoretical distribution had to be judged from the sample itself,the error which had been found by a few persons in that era was out of their depth,using maximum likelihood and representations in N-dimensional space and hypothesis test,focusing on the concept of degrees of freedom,Fisher＇s creative treatments not only corrected Pearson＇s error but completed the mathematical logic under the condition of parametric inference.So a well-rounded interpretation of degrees of freedom must consider these arguments adequately and we should strengthen the efforts on history of statistics as a result.