As is usual in the analysis of flood data, we assume that the initial distribution of x is the
exponential type:
F(x) = e
- x
It can then be shown that
Lim φn (x) = e -e -y^
n? 8
Where y is a transform of x such that
Y=an(x-un),
With 1/ an and un having the same dimension as x, so that y is a pure number.
Hence, the double exponential distribution e-e-y represents the cumulative probability
distribution of the reduced variety. The derivative of this gives the corresponding
probability distribution (probability density function) desired.
There are two possible approaches to the estimation of the theoretical distribution of the
largest value.
Firstly, if we know the functional form of the initial distribution and the values of its
parameters as well as the sample size n, then the parameters an and un, can be obtained
directly.
Secondly, if, as is more usual, the initial distribution is unknown, we can still estimate the
parameters an and un, provided we assume that the distribution is of the exponential
type. In this case, there are a number of alternative methods for estimating the
parameters an and un. We shall follow the large-sample modified least squares method
described by Gumbel ([2] pp. 35, 168-9).
As Gumbel shows, for large n, an and un can be estimated with a reasonable degree of
approximation independently of n. The estimates depend only on the sample distribution
of extreme values and are derived by the normal equations
1 Sx
---- = ------
a N
Yn
u = x - ------
a
Where Sx, x denote respectively the standard deviation and mean of observations of the
extreme values and N, Yn are functions of the number N of extreme values observed. The
values of an and Yn as functions of N have been tabulated by Gumbel ([2] p.228).