Kapetyn Distribution


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The 1962 National Bureau of Standards Report lists the Kapetyn distribution as another name for a generalized normal distribution.

Entry for Kapetyn distribution in [1].

There are sparse references to the Kapetyn distribution as a synonym for the “generalized” normal outside of the NBS report. The generalized family of normal distributions is a very large family of distributions that can model values that are normally distributed, right-skewed, or left-skewed relative to the normal distribution. Most references indicate that Kapetyn’s distribution refers to specific type of normal distribution; one plotted with a logarithmic horizontal scale (i.e., a log-normal distribution).

According to van der Kruit [1] Kapetyn proposed the following formula for the general skew probability density functions

Where F ′ is the derivative dF(x)/dx.

In particular, Kapetyn studied the formula

F(x) = (x +  κ)q

and the special case

q = -1,

which is the log-normal distribution.

Kapetyn Distribution History

Kapetyn’s wide system was refuted by Pearson on several grounds [3], including that it was too general and that the lognormal curve used by Kapetyn was of limited skewness in some cases. However, it appears that Pearson misunderstood Kapetyn’s theory.

Although “Kapetyn’s distribution” as a name is lost to the annals of history, his formula inspires the works of many authors including Edgeworth [4], Wicksell [5], and Gilbrat [6].

References

[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] van der Kruit, P. (2014). Jacobus Cornelius Kapteyn

Born Investigator of the Heavens. Springer International Publishing.

[3] Aitchison, J. & Brown, J. (1963). The Lognormal Distribution. Cambridge at the University Press.

[4] Edgeworth, F. Y. (1898). On the representation of statistics by mathematical formulae. Journal of the Royal Society 1, pp. 670‒700.Google Scholar

[5] Wicksell, S.D. (1917). On logarithmic correlation with an application to the distribution of ages at first marriage. Medd. Lunds. Atr. Obs. 84.

[6] Gilbrat, R. (1930). Une loi des repartitions economiqeus: l’effet proportionnel. Bull statist. Gen. Fr. 19, 469.