Another characterization of the type I extreme value distribution

Statistics & Probability Letters 5 (1987) 83-85 North-Holland

March 1987

ANOTHER CHARACTERIZATION OF THE TYPE I EXTREME VALUE DISTRIBUTION

Rocco BALLERINI University of Florida, Gainesville, F L 32611, USA

Received July 1986 Revised September 1986

Abstract: The linear-drift Gumbel record (ldGr) model is defined as Y. = X n + cn, n = 1, 2 . . . . . and c > 0 where { X. } is i.i.d.

with type I extreme value distribution A..p. A characterization of A~.O is given by showing that the ldGr model is the only model of the form Y. = X n + cn yielding M. = max{ Y1, }I2..... Yn} independent of the indicator of a record at time n. Keywords: records, Gumbel distribution, characterization.

I. Introduction

In two recent papers, Ballerini and Resnick (1985, 1986a) have examined (somewhat extensively) the occurrence of upper records from the simple linear regression model

Y.=X.+cn,

n=l,2 ..... c>0,

(1.1)

where { X. }.~-1 is strictly stationary satisfying certain moment assumptions and mixing conditions. In the special case when { 3(. } is i.i.d, with type I extreme value (or Gumbel) distribution given by a ~ , B ( x ) : = exp{ - e x p { - 13-1< X - oo < x < ~ ,

--

Or)} ) , (1.2)

(1.1) is referred to as the linear-drift Gumbel record (ldGr) model. In a third paper, Ballerini and Resnick (1986b) show that for the ldGr model the random variable M. = max(Y x, Y2. . . . . Y.} is independent of the indicator 1, .'= l l r e c o r d o c c u r s a t t i m e n ] = l [ y . > M._~] for each n. This somewhat unexpected result follows directly from the fact that for the ldGr model, the

OO sequence { M . }.-1 can be embedded in an extremal process. The reader is referred to the latter paper for details. A partial converse to this result is proved here, showing that independence of M. and 1. for models of the form (1.1) and {X.} i.i.d, holds only when the Xi's have a Gumbel distribution. This characterization emphasizes the importance of the ldGr model in record analysis. The most celebrated characterization of A~,~ appears within the framework of extreme value theory, stating that A,,t~ is the only max-stable probability distribution function with the entire real line as its support (see e.g. Theorem 1.4.1 of Leadbetter, Lindgren, and Rootzdn (1983)). There are also numerous characterizations of not just A,,a itself, but of the maximal domain of attraction of A.,0. For these the reader is referred to de Haan (1970). Before proceeding with the statement and proof of the theorem, we need to make several observations. For ( X, } i.i.d, with distribution function F, " 1F (x + ci) and note that let G,(x; c) := Fli_

G,(x; c)$Goo(x; c ) , = f i F ( x + c i )

asn~oo.

i--1

This research was partially supported by a Research Development Award from the University of Florida.

Since Goo(x; c) = P(V~° I( Xi - ci) <~x) where "V"

0167-7152/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

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STATISTICS & P R O B A B ILITY LETTERS

denotes the maxiriaum, a simple application of the Borel-Caatelli lemma reveals that Goo is a nondefective p r o b a b i l i t y distribution f u n c t i o n whenever EX~ < oo. Now let

p. := E[1.] : f?oo G,,(x;

c)F(dx)

-- 00

Goo(x; c ) F ( d x ) =

get

P(M,,-cn<~x, 1,=1)

=p.P(M. - cn ~ x) =p.r(x)G._,(x; c). (2.2) Let x 0 = i n f { x ; F ( x ) > 0} and x 1 = sup{x; F ( x ) < 1} denote the left and right endpoints of F, respectively, and H , ( y ) := F(y)Gn_x(y; c). F r o m (2.1) and (2.2) we have

and

p:=p(c)=

March 1987

limp.

n ~ O0

LSan_a(y;

(by monotone convergence). Observe that as a function of c, p ( c ) is a distribution function since p(c),~O as e J.0, p ( c ) l ' l as e l ' r e .

c)F(d y) = pnF(x)Gn_l(X;

c),

X>Xo,

and both sides vanish when x ~
f~IH.(y)F(Y)-IF(dy)

2. A characterization of A,,,# Theorem. Suppose { X, } is i.i.d, with nondegenerate, continuous c.d.f. F and EX~ < oo. I f for every c > O, M n is independent of 1, for each n, then F = A ~,~ for some real a and fl > O.

=p.H,,(x),

x > x o. (2.3)

If Q ( x ) := - l o g F(x) then the continuity of F allows (2.3) to be rewritten as

L~H,,(Y)(-Q(d y))= p , , H . ( x ) .

Proof. First note that n-1

N o w integrating H ~ l ( y ) over (z, oo) with respect to p,,H.(y) yields

P ( M , - cn < x ) = I-I F ( x + ci) i=0

and

~=H.(y)-lp.H.(dy)

P(m.-cn
1.=1)

=Sz°°H.(y)-ld[fYxoH.(x)(-Q(dx)) ] oo 1

=P(~(Xi+ci)<~x+cn, i

.-1

= f H.(y)- H,,(y)(-Q(dy))

))

X,,+cn> V ( X i + c i

= foo(_ Q(dy)) = -log

i--1

r(z).

n-1

=P

V (x._,-ci)<=.x,

But f ~ H . ( y ) - l p . H . ( d y ) =

-p,, log H,,(z) so

i--O

- l o g F ( z ) = - p . log H,,(z).

"-1 )) X.> V ( X , , - i - c i

Taking limits as n ---) oo on both sides of (2.4) and letting H ( z ) = lq•=oF(z + ci) we get the relationship

i--1

=?

tv

(x._,-ci)<.X.~x

\i=1

= Lx G . _ i ( y ; c ) F ( d y ) .

(2.4)

)

H ( z ) = r ( z ) lip

(2.1)

for all real z.

(2.5)

Replace z. by y - cn in (2.5) for some real y to get oo

Since M, is independent of 1, by hypothesis, we

F ( y - cn) '/p = H ( y - cn) = I-I F ( y - cn + ci). i~O

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STATISTICS & PROBABILITY LETTERS

L e t t i n g n = 1, F ( y

-

c) 1/p

=

F ( y - c ) F ( y ) alp a n d

hence

F(y-c)=

or

F(y)=exp(-Q(O)e F

(y)

(1 _ p ) - t

for all real y a n d

c>0.

S i n c e p ( c ) is a d i s t r i b u t i o n f u n c t i o n , let O ( c ) = ( 1 - p ( c ) ) -1 so t h a t l ~ < 0 ( c ) < ~ , 0 ( c ) $ 1 as c $ 0 a n d O ( c ) ? o o as c $ o o . W e t h u s h a v e the

(2.6)

N o w s u p p o s e x o > - oo. F o r f i x e d c > 0, c h o o s e y > x 0 s u c h t h a t y - c ~< x 0. By the d e f i n i t i o n o f x 0 a n d the fact t h a t O(c) >1 1, F ( y ) ° 0 b u t F ( y - c) = 0 w h i c h c o n t r a d i c t s (2.6). H e n c e , x o = - oo. S i m i l a r l y it f o l l o w s that x 1 = + oo. F o r p o s i tive real n u m b e r s c 1 a n d c 2, (2.6) y i e l d s

F ( y - c 1 - c 2) = F ( y ) °(¢1+~2). Also

F ( y - c I - c2) = r ( y - c l ) °t¢2) = F ( y ) °(c')°(c2) w h i c h i m p l i e s O(c 1 + c2) = O(cl)O(ct). T h i s s h o w s O(c) = e ~ for s o m e c o n s t a n t a. L e t t i n g y = c in (2.6) yields

F(c) = r ( 0 ) °-°'

Therefore, []

F=A.,¢

-~y),

-oo
for s o m e real a a n d fl > O.

Acknowledgements

relationship

r ( y - c) = V ( y ) °re).

March 1987

I gratefully acknowledge my advisor Professor S.I. R e s n i c k for his g u i d a n c e in t h e thesis f r o m w h i c h this r e s e a r c h is a part.

References

Ballerini, R. and S.I. Resnick (1985), Records from improving populations, J. Appl. Prob. 22, 487-502. Ballerini, R. and S.I. Resnick (1986a), Records in the presence of a linear trend, Adv. AppL Prob., to appear Dec. 1987. Ballerini, R. and S.I. Resnick (1986b), Embedding sequences of successive maxima in extremal processes, J. Appl. Prob., to appear. Haan, L. de (1970), On Regular Variation and its Application to the Weak Convergence of Sample Extremes, MC Tract 32 (Mathematisch Centrum, Amsterdam). Leadbetter, M.R., G. Lindgren and H. Rootz@n (1983), Ex-

tremes and Related Properties of Random Sequences and Processes (Springer-Verlag, New York).