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Biometrika 1958 45(3-4):447-470; doi:10.1093/biomet/45.3-4.447
© 1958 by Biometrika Trust
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APPROXIMATE FORMULAE FOR THE STATISTICAL DISTRIBUTIONS OF EXTREME VALUES

J. J. DRONKERS

Ruychrocklaan 180, The Hague

This paper deals with the distribution function of the order statistics

xn–m+1(m= 1, 2...), Mn, m(x).

For this distribution function of the mth largest value (if m is counted from above) approximate formulae are derived.

These formulae are generalizations of the corresponding approximate formulae for the distribution of the extremes proper. Successively the initial distributions f(x) are supposed to be of exponential type, Cauchy type or of limited type (finite range).

We first deal with the basic conditions to be imposed on the initial distributions f(x). An expansion formula has been derived for the distribution of the excess, 1 – F(x), which plays an important part in the investigations of this paper.

We then consider the general formula of the initial distribution of the mth values, Mn, m(x). Appropriate formulae have been derived to determine the mode and the maximum value of Mn, m(x). The behaviour of the maximum value by varying n and m has also been studied and approximate formulae for Mn, m(x) are successively deduced. Every succeeding formula has a more restricted range of application. Finally, limiting expressions for Mn, m(x) are deduced for the three types of initial distributions, mentioned above. The well-known limiting functions of Fisher—Tippett and Gumbel are deduced again. Formulae for the determination of the interval of application have been deduced.


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