5 Some results of the statistical theory of extreme values
5.4 Maximum Domains of Attraction
: (5.6a)
When using that formala twice for numberssandtone finds the functional equations
a(st) = a(s)a(t); b(st) = a(s)b(t)+b(s): (5.6b) Eqs. (5.6) determines the (non–degenerate) max–limit distributions for maxima. For example seta(t) 1. Then fromb(st) = b(t)+b(s)it follows that b(t) = lnt1=. Without loss of generality we may writeP(x)=e-f(x), leading totf(x)=f(x+lnt1=) and thus finally tof(x) = e-x with > 0 and arbitrary . Comparision with eq. (5.5c) shows that these distribution functions belong to the type of the GUMBEL distribution. The norming constants for the three standard max–limit distributions can be read off fromPN(cNx+dN)=P(x)to be We note that the three max–limit laws of the FISHER–TIPPETT theorem are tightly connected for a positive random variableX,
(x) = (lnx) = (-1=x): (5.8) This means that ifX has a distribution function , the random variable lnX will have a distribution functionand the random variable-1=Xwill have a distribution function .
5.4 Maximum Domains of Attraction
We define themaximum domain of attraction (MDA)of a max–limit distributionP as the set of all distribution functionsPfor which eq. (5.3) holds for appropriate real numberscN>0anddN. SincecNx+dN! xmaxforN!1(as is seen from eq. (5.3)) andP(x)!1forx!xmax(norming condition) this is equivalent to
lim
5.4 Maximum Domains of Attraction From the convergence to types theorem it is clear that different norming constants
c
N, dNfor the same distribution function Pcan not lead to convergence to different standard max–limit distributions. One can see from that formula furthermore that the behavior ofPnear its right endpointxmax(which may be1) is important for the affiliation ofPto MDA(P).
Next we state the definitions of tail–equivalence and regular variation that will be useful in characterizing the MDAs of the standard max–limit distributions. We define two distribution functionsP1andP2to betail–equivalentif they have the same right endpointxmaxand if
lim
x!xmax
1-P
1 (x)
1-P
2 (x)
= c with 0 < c < 1: (5.10) One is able to show for each of the three standard max–limit distributionsPthat being tail–equivalent is a sufficient condition for any two distributions to be a member of the same MDA(P). Furthermore one is able to specify a relation between the norming constants of the two distributions in this case. This feature is called “closure property”.
For details about any of the three standard max–limit distributions see the subsections below.
We define a positive, LEBESGUEmeasurable functionf(x)on(0;1)to be regularly varying at point1of indexif for eacht>0
lim
x!1 f(tx)
f(x)
= t
: (5.11)
5.4.1 MDA of the FRECHET´ distribution
Since the FRECHET´ distribution has an algebraic tail,
1-
(x) x
- for x ! 1; (5.12)
one expects P 2 MDA() to be power–tailed. Indeed the MDA of the FRECHET´ distribution can be easily characterized by regular variation of the distribution tails.
We state without proof that a distributionPis in the MDA of the FRECHET´ distribution
if and only if its tail is regularly varying of index, i. e.
lim
x!1
1-P(tx)
1-P(x)
= t
for each t > 0:
Note that all distributions in MDA() have a right endpoint xmax = 1 and that
d
N
= 0 can always be chosen. Another characterization of MDA() can be given whenPis an absolutely continuous distribution function with densityp(x)=P0(x). It is a sufficient condition forPto satisfy theVONMISES condition
lim
x!1
xp(x)
1-P(x)
= > 0 (5.13)
in order to be in MDA(). Furthermore it holds true that MDA() consists of all distribution functions obeying theVON MISES condition eq. (5.13) and their tail–
equivalent distribution functions. Distributions of this type are e. g. PARETO–like dis-tributions and the CAUCHYdistribution.
For calculating norming constants it is usefull to state the closure property of MDA(): LetP1andP2be distribution functions and letP12MDA()with norm-ing constantscN>0, that means for eachx>0
lim
N!1 P
N
1 (c
N
x) =
(x): (5.14a)
Then
lim
N!1 P
N
2 (c
N
x) =
(ax) (5.14b)
if and only if
lim
x!1 1-P
1 (x)
1-P
2 (x)
= a
: (5.14c)
In the Quenched Jump Model of chapter 6 we use the fact that the density function
()=
-1- of the trapping time =exp(-E) is of the PARETO type and so the FRECHET´ max–limit distribution applies. Its tail
Z
1
d
0
( 0
) =
-is equivalent to the asymptotic tail of the FRECHET´ distribution
1-
()
- for !1:
From the closure property eq. (5.14) one can deduce that the norming constants of
itself apply, which are given by eq. (5.7a). Thus the maximal value ofNrandom trapping timesTiscales for largeNas
max(N) cN = N1=: (5.15)
This behavior can be understood more easily from the following simple argument:
The typicalmax(N) should be determined by the condition that among N random variables there is on average one equal or larger thanmax(N). Thus the tail should scale asN-1, yielding
prob(Tmax) -max(N) N
-1
() max(N) N1=:
5.4 Maximum Domains of Attraction 5.4.2 MDA of the WEIBULL distribution
Without proof we state that a distribution functionPis in the MDA of the WEIBULL distribution if and only if it has a finite right endpointxmax < 1and if the function
P
?
(x)=1-P(xmax-x-1)is regularly varying of index, i. e. if for anyt>0 lim
x!1
1-P xmax-(tx)-1
1-P xmax-x-1
= t
:
As in the case of the FRECHET´ distribution this condition can be reformulated with a
VON MISES condition. LetP be an absolutely continuous distribution function with densityp(x)=P0(x)which is positive on some open finite interval(z;xmax). If
lim
x!xmax
(xmax-x)p(x)
1-P(x)
= > 0; (5.16)
then P 2 MDA( ). Furthermore MDA( ) consists of all distribution functions satisfying theVONMISES condition eq. (5.16) and their tail–equivalent distributions.
The WEIBULLdistribution possesses a closure property: LetP1andP2be distribu-tion funcdistribu-tions and letP12MDA( )with norming constantscN >0, that means for eachx<0
lim
N!1 P
N
1 (c
N
x+xmax) = (x): (5.17a)
Then
lim
N!1 P
N
2 (c
N
x+xmax) = (ax) (5.17b)
if and only if
lim
x!xmax
1-P
1 (x)
1-P
2 (x)
= a
-: (5.17c)
5.4.3 MDA of the GUMBELdistribution
The tail of the GUMBEL distribution decays exponentially,
1-(x) e
-x for x ! 1;
so one might expect all exponentially tailed distributions to belong to MDA(). This is right, but as it turns out, the maximum domain of attraction of this max–limit distribution is much larger. Hence MDA() is more difficult to characterize than MDA()and MDA( ).
When stating it memorable though not quite exact the MDA of the GUMBEL dis-tribution contains all disdis-tribution functions whose tail decay much faster than any al-gebraic law. “Much faster” decaying distributions may contain any whose tail decays
as a lognormal distribution or faster. Thus the importance of the GUMBELdistribution can be compared to the importance of the GAUSSIAN distribution as a limit distribu-tion for sums of random variables.
For the sake of completeness we give a characterization of the MDA of the GUMBEL distribution:P2MDA()if and only if there exists a positive functiona(t)such that for each realt
lim
x!xmax
1-P x+ta(x)
1-P(x)
= e -t
: (5.18)
A possible choice fora(t)is
a(x) = 1
1-P(x) xmaxZ
x
dt 1-P(t)
: (5.19)
Also the GUMBEL distribution has a closure property that can be usefull when computing norming constants: LetP1andP2be distribution functions with same right endpointxmax and letP1 2 MDA() with norming constants cN > 0 anddN, that means for each realx
lim
N!1 P
N
1 (c
N x+d
N
) = (x): (5.20a)
Then
lim
N!1 P
N
2 (c
N x+d
N
) = (x+a) (5.20b)
if and only if
lim
x!xmax
1-P
1 (x)
1-P
2 (x)
= e a
: (5.20c)