A graphical method for the analysis of statistical distributions into two normal components
A graphical method for the analysis of statistical distributions into two normal components
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用于分析两个正态分量的统计分布的图形方法
DOI：
10.1093/biomet/40.3-4.460
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发表时间：
1953
期刊：
影响因子：
2.7
通讯作者：
E. J. Preston
中科院分区：
文献类型：
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作者：
E. J. Preston
在统计实践中出现的许多频率分布，可能是由于在考虑的宇宙中存在两个或多个独立的子宇宙，每个子宇宙都近似为正态；例如，英国男人和女人的身高或体重，或者专业工程师和工匠工程师的智商，都可能具有这种属性。如果子宇宙有不同的平均值，不同的方差，包含不同的个体数，就会产生各种各样的复合分布。从频率曲线的双峰或多峰外观来看，子宇宙的存在通常并不明显，因为除非平均值的分离相当大，否则不会出现单独的模式；如果它们的尺寸相当，则需要大约是组件标准偏差的三倍。因此，下面的问题出现了：给定一个样本（最好是几千个人），从一个由于某种原因被怀疑具有这些成分的宇宙中选择出来，我们能确定子宇宙最可能的性质吗？也许最重要的是，有没有一种快速实用的方法来做到这一点？“矩量法”最初是由皮尔逊（K. Pearson, 1894）在理论上解决这个问题时使用的。从那时起，更有效的“最大似然法”被RA Fisher和其他人开发出来，并被CR Rao（1948）特别应用于这个问题。在同一篇论文中，Rao还通过矩量法给出了一个快速而优雅的理论解，用于假设两个分量具有相等方差的情况。这取决于三次方程的解，并且似乎产生相当精确的结果。然而，在我们看来，所有这些贡献都存在着过于复杂和需要进行冗长计算的问题。这些方法确实能从现有的样本中得出最准确的结果，但考虑到抽样波动和假设真实成分是正态的且方差相等所造成的不可避免的固有误差，这种程度的准确性似乎是不必要的。因此，似乎有空间采用更快速、更省力的图形化方法。
Many frequency distributions occur in statistical practice which are probably due to the existence of two or more separate sub-universes within the universe under consideration, each of which is of approximately Normal form; for instance, the heights or weights of English men and women, or the intelligence quotients of professional and artisan engineers would probably have this property. If the sub-universes have different mean values, different variances and contain different numbers of individuals, a great variety of composite distributions will result. The existence of sub-universes is not usually obvious from the bimodal or multi-modal appearance of a frequency curve, since separate modes do not appear unless the separation of the means is considerable; it needs to be about three times the standard deviation of the components, if they are of comparable size. The following problem therefore arises: given a sample (preferably of several thousand individuals), selected from a universe suspected for some reason of having such components, can we determine the most probable nature of the sub-universes? Perhaps most important, is there a quick practical way of doing so? The'method of moments' was first used in a theoretical solution of the problem by K. Pearson (1894). Since then, the more efficient'method of maximum likelihood'has been developed by RA Fisher and others, and applied in particular to this problem by CR Rao (1948). Rao also gives, in the same paper, a rapid and elegant theoretical solution, by the method of moments, for the case of two components assumed to have equal variances. This depends upon the solution of a cubic equation, and appears to yield quite accurate results.However, it seems to us that all these contributions suffer from over-complexity and the necessity for lengthy calculations. The methods do indeed give the most accurate results possible from the available sample, but this degree of accuracy seems rather unnecessary in view of the unavoidable inherent errors due to sampling fluctuations and to the assumptions that the true components are normal and have equal variances. Thus there seems to be room for a more rapid, and much less laborious, graphical method.