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Kelley should be spelled Kelly. Moreover, I think defining it before the more general version of skewness makes more sense. |
→Quantile-based measures: correction |
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A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984):<ref name=Groeneveld1984>{{Cite journal | doi = 10.2307/2987742 | last1 = Groeneveld | first1 = R.A. | last2 = Meeden | first2 = G. | year = 1984 | title = Measuring Skewness and Kurtosis | jstor = 2987742| journal = The Statistician | volume = 33 | issue = 4| pages = 391–399 }}
</ref><ref name=MacGillivray1992>MacGillivray (1992)</ref><ref name=Hinkley1975>Hinkley DV (1975) "On power transformations to symmetry", ''[[Biometrika]], 62, 101–111</ref>
:<math> \gamma( u )= \frac{ Q( u ) +
This leads to a corresponding overall measure of skewness<ref name=MacGillivray1992>MacGillivray (1992)</ref> defined as the [[supremum]] of this over the range 1/2 ≤ ''u'' < 1. Another measure can be obtained by integrating the numerator and denominator of this expression.<ref name=Groeneveld1984/> The function ''γ''(''u'') satisfies −1 ≤ ''γ''(''u'') ≤ 1 and is well defined without requiring the existence of any moments of the distribution.<ref name=Groeneveld1984/> Bowley's measure of skewness is γ(''u'') evaluated at ''u'' = 3/4 while Kelly's measure of skewness is γ(''u'') evaluated at ''u''=0.1.
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