対称準範囲及び準範囲中央について

The statistical theory of extreme values was first systematically investigated in about 1925 by L. Von. Borpiewicz [1] and L. H. C. Tippett [2] for analizing the problem of flood especially. After that it has been developed by R. Von. Mises [8], Cumbel [3] [4] [7] and others, and this theory has become to be used in many different fields where the problems related to the extreme values appear, for instance, in quality control, in economics and in connection with breaking strength of material.
The statistcal studies of extreme values are meant to give an answer for two types of question :
(1) Does an individual observation in a sample taken from a distribution, alleged to be known, fall outside, what proportion may reasonably be expected ?
(2) How we may use the extreme statistics for the theories of testing hypothesis and estimation ?
In this paper we shall investigate some properties of the generalized range and midrange for the second purpose, that is, we shall first derive the asymptotic distribution of mth-range and mth-midrange under the initial symmetric exponential type and second we shall compute the exact distribution for small sample size under the standarized normal distribution.