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What is the generalized normal distribution?
The generalized normal distribution, sometimes called the generalized error distribution or Kapetyn distribution, is a symmetric distribution family commonly used in mathematical modeling. They are particularly useful when errors — the difference between expected and observed values — do not follow a normal distribution. The family includes the normal and Laplace distributions as special cases.
The generalized normal distribution is especially useful when investigating errors around the mean or in the tails (extremes). Alternative distribution families can also be used to investigate errors. For instance, both the t-distribution and Cauchy distribution have heavier tails, which means they are more likely to produce extreme values than the normal distribution. Which distribution you use for analysis depends on properties of the data, the goals of the analysis, and the specific application. For example, if you have a small sample size you may want to choose the t-distribution as it is more robust to small samples than the generalized normal distribution.
Properties of the generalized normal distribution
Three parameters define the generalized normal distribution:
- The mean, μ: the mode (peak) of the distribution. The median and mode are equal to μ, similar to the standard normal distribution.
- The standard deviation, σ: the data’s dispersion around the mean.
- A shape parameter, Β (kurtosis), indicates the amount of data in the tails.
A generalized normal distribution with Β = ½ equals the normal distribution, while Β = 1 equals the double exponential (Laplace) distribution. When Β tends toward zero, the distribution resembles a uniform distribution.
The Generalized Error Distributions (GED) have two classes:
- First kind (GED-1): These distributions have heavy tails.
- Second kind (GED-2): These distributions have highly skewed tails.
The Kapetyn distribution
The 1962 National Bureau of Standards Report lists the Kapetyn distribution as another name for the generalized normal distribution.
There are sparse references to the Kapetyn distribution as a synonym for the “generalized” normal outside of the NBS report. Kapetyn’s distribution sometimes refers to a 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 Karl 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 may have misunderstood Kapetyn’s theory.
Although “Kapetyn’s distribution” as a name is lost to the annals of history, the formula inspired 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.