Geoscience Reference
In-Depth Information
large value of
x
, denoted by
u
n
,or
u
n
)
2
2!
u
n
)
3
3!
f
(
u
n
)
(
x
−
f
(
u
n
)
(
x
−
F
(
x
)
=
F
(
u
n
)
+
f
(
u
n
)(
x
−
u
n
)
+
+
+···
(13.47)
From the definition of exponential type distributions (13.45) for large values it follows
that
[
f
(
u
n
)]
2
[
f
(
u
n
)]
3
=
−
+
f
(
u
n
)
F
(
u
n
)
;
f
(
u
n
)
=
F
(
u
n
)]
2
; etc.
(13.48)
1
−
[1
−
The value of
u
n
is fairly arbitrary, but if it is defined such that its probability is given
by
F
(
u
n
)
the derivation becomes especially straightforward. (Note that this
is almost, but not quite, the average probability of the largest event in the sample,
namely 1
=
1
−
1
/
n
,
1) shown in Equation (13.19)). After substitution of (13.48) and
some algebra, (13.47) can be written as
−
1
/
(
n
+
F
(
u
n
)]
1
n
(
x
u
n
)
2
n
(
x
u
n
)
3
+
α
−
−
α
−
F
(
x
)
=
1
−
[1
−
−
α
n
(
x
−
u
n
)
+···
2!
3!
in which, by definition
F
(
u
n
)], which can be considered a constant
parameter; thus, with the definition of
u
n
, this series becomes
α
n
=
f
(
u
n
)
/
[1
−
1
n
F
(
x
)
=
1
−
exp[
−
α
n
(
x
−
u
n
)]
(13.49)
Recall that
F
(
x
) is the initial distribution of the population from which the
n
items of
the sample were taken, and that it indicates the probability that any item of the sample
is smaller than or equal to
x
. The probability that all
n
items are smaller than or equal to
x
,is
[
F
(
x
)]
n
G
(
x
)
=
(13.50)
provided the items are independent of one another. Combination of Equations (13.49)
and (13.50) produces
1
u
n
)]
n
1
n
G
(
x
)
=
−
exp[
−
α
n
(
x
−
(13.51)
In the limit, as the size of the sample is allowed to increase indefinitely, so that
n
,
one obtains (Abramowitz and Stegun, 1964, 4.2.21) the asymptotic distribution function
of the largest values
→∞
G
(
x
)
=
exp[
−
exp(
−
y
)]
(13.52)
G
(
x
), or
and the corresponding density function
g
(
x
)
=
g
(
x
)
=
α
n
exp[
−
y
−
exp(
−
y
)]
(13.53)
where
y
u
n
) is the reduced largest value. The distribution and the density func-
tion of the extremes are denoted in this section by
G
(
x
) and
g
(
x
), merely to distinguish
them from the initial distribution function and the initial density function, respectively.
=
α
n
(
x
−
