Elfving distribution

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The Elfving distribution is defined as [1]

elfving distribution

The distribution is named after Finnish statistician and mathematician Gustav Elfving (1908-1984), who described the distribution in 1947 [2].

In his Biometrika paper, Elfving established a distributional result concerning order statistics. Specifically, he investigates the distribution of the sample range when the samples are drawn from a standard normal distribution. Previous work on exact calculations for finite samples were found to be intractable [3]. Elfving tackled the problem from a different direction, determining the asymptotic distribution of the sample range for large samples.

Elfving Distribution Alternatives

Elfving’s distribution has a distinct disadvantage over other methods, which may be why it’s little discussed outside of a few historical references. While other methods can be expressed directly in terms of the range, Elfving’s formula involves a non-linear transformation of the range, making it a theoretical challenge [4]. For example, Gumbel’s method [5] leads to the same results as Elfving’s distribution; however, Gumbel’s method requires no knowledge of sample size, the analytical form of the initial distribution, or numerical values of the distribution’s parameters.


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

[2] Elfing, G. (1947). The asymptotical distribution of range in samples from a normal population. Biometrika 34 111–119.

[3] Nordstrom, K. (1999). The Life and Work of Gustav Elfving. Statistical Science. Vol. 14, No. 2, 174-196.

[4] Cox, D. R. (1948). A Note on the Asymptotic Distribution of Range. Biometrika35(3/4), 310–315. https://doi.org/10.2307/2332353

[5] Gumbel, E. (1949). Probability tables for the range. Biometrika, 36: 142-148.