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CENTER FOR STOCHASTIC PROCESSESN

, Department of StatisticsUniversity of Nort CarolinaChapel Hill, North Carolina

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A BROWNIAN BRIDGE CONNECTED WITH EXTREME VALUES

by

L. do Haan

DTICS ELECTEDEC 0 9 8

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Technical Report No. 241

September 1988

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11. TITLE (include Security Ciaa'ailc"tOn)A Brownian bridge connected with extreme values

12. PERSONAL AUTHOR(S) de Haan, L.

13&. TYPE OP REPORT 131L TIME COVERED 14d. DATE FRPR Y~ o.Dy i.PG ONpreprint FROM 1/9/88 To 1 '8/89 l88 September I 1

IS SUPPLEMENTARY NOTATION

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17. COSATI CODES IBt SUBJECT TERMS (Continue on reperee if necemgry and identify by btoch nunmber)FIELD GROUP IsuB. GR. Key words and phrases: N/A

1S. ABSTRACT tContinue an oeuerae if necessary and idenify by btock number)

A stochastic process formed from the intermediate order statistic is shown to

converge to a Brownian bridge under conditions that strengthen the domain of

attraction conditions for extreme-value distributions.

DO FORM 1473.83 APR EDITION OF I JAN731 IS -OWLAlE. INL-TT~

-1-

A Brownian bridge connected with extreme values

L. de Haan

Erasmus University Rotterdamand

Center for Stochastic ProcessesUniversity of North Carolina

Abstract

A stochastic process formed from the intermediate order statistic is shown to

converge to a Brownian bridge under conditions that strenghten the domain of

attraction conditions for extreme-value distributions..

1. Introduction r#Suppose X1 , X2 , ... are i.i.d. random variables from some distribution

function F. Let X(1,n ) < .< X(n,n ) be the n-th order statistic.

It is well-known (cf. e.g. Resnick 1987) that, if F is the domain of

attraction of an extreme-value distribution G the point process represented

by the points

{ .-b}Ooa, I n )4 i =1

(with a > 0 and b chosen appropriately, n = 1, 2, ... ) converges (n-*o) inn n dG (x)

distribution to a Poisson process with mean measure dt.-5.(x. From this

convergence one can infer the joint limiting distribution of extreme order

statistics (X(nn) *.., X(n-k,n)) with k fixed (n-po). However limiting

distributions for intermediate order statistics X(n.k,n) where k = k(n) -

k(n)/n -* 0 (n-vo) cannot be inferred from the above-mentioned point process

convergence.

If F(x) = 1 - e -x (x > 0), then by R~nyi's representation X(n.k,n) and the

stochastic process {X(n-k+fksl,n) - X(nk,n)}0s<l are independent. Since the

latter is' equal in distribution to {X( ] k)}041 it converges - if

Research supported by AFOSR F49620 85C 0144 at the Center for stochastic

processes, Chapel Hill N.C.

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properly normalized - to a Brownian bridge, provided k = k(n) -* 00 (n-I-o). We

are going to extend this behaviour to any distribution function satisfying a

natural strengthening of the conditions for the domain of attraction of an

extreme-value distribution.

So, in contrast with the situation for extreme order statistics, the process

convergence underlying the limit behaviour of intermediate order statistics,

is Gaussian. The same phenomenon by the way is present in the neighborhood of

a quantile of the distribution (compare the asymptotic normality of sample

quantiles with the point process convergence of de Haan and Resnick (1981)).

2. The results

Theorem 1

Define U:= (1_ ) (inverse function) and suppose U has a positive derivative

U'. Suppose further that U r RV and there exists a positive function a suchTthat for some p ; 0 and all x > 0 (with either choice of sign)

(2.1) lim= (tx)l-1 u'(tx) - t-(U'(t) ±x - 1t( M= a(t) -p

(read ± log x for the right-hand side if p = 0), then for k = k(n) -* 00,

k(n) = o(n/g"(n)) where g(t):= t 3 - 2 r{U'(t)/a(t)}2 , n-o

1. there exists a sequence of Brownian bridges Bn(s) such that for all e > 0

(2.2) sup V1.s T+14 X(n-[ks],n) X(n-kn) + B--B(s)0<8<1 [{1 - F(X(n-k,n))}/F'(Xn_k,n)) n

in probability (n-to).

2.

(2.3) 1 (n-kn) U(J) is asymptotically standard normal.n

i .i._ .. . -- w=,~w ~ im.mm m nu nlommlmmi U , - ,

-3-

3. the process under 1. and the random variable under 2. are asymptotically

independent.

Before proceeding to the proof of the theorem we first formulate condition

(2.1) in terms of the distribution function and its density (for the proof of

the equivalence see Dekkers and de Haan 1988, section 2 and appendix).

Proposition 2.1

Relation (2.1) with p = 0 is equivalent to:

for y > 0: ±tl+l/F'(t) e 1(b),

for -f < 0: U(oo):= lim U(t) < o and ;t-1 -l/fF'(U(o)-t -1 ) c= 1(b),

for y = 0: let f0 = (1-F)/F' and x := supfxlF(x) < 1}. There exists a positive0*

function c with ot(t) -* 0 (ttx) such that for x > 0 locally uniformly

1-F(t+xf 0 (t)) X 2

lim 1-F(t) e + rettx o (t)

Proposition 2.2

Relation (2.1) with p > 0 is equivalent to:

for y > 0: for some c > 0 the function tl+lI'F'(t) - c is of constant sign and

lir (xt)l+l/rF'(tx) - c = x-p

t-oo tl+l/TF'(t) - c

(regular variation with exponent -p, notation t{t1+l/TF'(t) - c} -e RV_p).

for T < 0: for some c > 0 the function tt--'/TF'(U(oo) - t-1 ) - c} e RVp

ForThe proof of the theorem will be broken up in a series of lemma's.

Lemma 1 Just, f:cnlc _ 0

Relation (2.1) implies DIM'.copy' R -. _

NSPCTE DIstributIon ..

(2.4) U'eRV S AvallablItv Cc. s

A- an/rot~qt Special

10

= , _ . . . . . , I - ,I mm m mm m • ml m "

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(i.e. U' is regularly varying at infinity with index y-1) and

(2.5) lim { U(tx) -U(t) xT- 1} /al(t) = %pl(x)

for x > 0 locally uniformly, where ao and a1 are positive functions and

(2.6) Y(X) -- 0

(( (log x) 2/2 T =0.

Furthermore

- U(t) >0

(2.7) ao(t) t U'(t) y=0

-({U(oo) - U(t)} < o

and

T-1a(t) ty/U(t) 'Y > 0

(2.8) al(t) a(t)/ft.U'(t)I 7=0

L(-Y)-l a(t ) ty/AU(0o) - UMt) T" < 0.

Corollary 1

a0 c RV and aI 1 RVO. Moreover lim a1 (t) = 0.t-Woo

Corollary 2

Relation (2.5) also holds with ao(t) = t U'(t) for Pll y (cf. Dekkers and de

Haan, th. Al and A3).

Proof

>0. Relation (2.1) implies (cf. Dekkers and de Haan 1988, th. A.1)

L .

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(2.9) (tx)-7. U(tx) - t-( U(t) x - 1

urn-1 - Pt- o "- . a(t )

locally uniformly for x > 0, i.e.

rU(ta _ U(t) "tU-P

(2.10) Jim 'x UM T fy- / a(t) t/U(t)} t X . -

0: Relation (2.1) implies (cf. Dekkers and de Haan 1988, th. A.5)

(2.11) lim {U(tx) -U(t) - log x/ {a(t)/t.U'(t)} =_± (log x)2/2t-*W t U'(t)

for x > 0.

y < 0: Relation (2.1) implies (cf. Dekkers and de Haan 1988, th. A.3)

(2.12) lim (tx)-T{U(o0) - U( tx)} - t-{.fU(oo) - U(t)} x- p - 1-1 -tOM -1 .a(t) -

for x > 0 locally uniformly, i.e.

(2.13) Jim {U(tx) - U(t) x- 1}/{(_)-la(t)tT/(U(oo)_U(t))}(2.13 lia -TiU(0o ) - U(t)} l

x-T x - p - 1-p

for x > 0.

Lemma 2

If (2.1) holds with a + sign for -y Z 0 and a - sign for 7' < 0, then given

c > 0 there exists t0 such that for t t0 and x > 1

1. T > 0

(l-e, ,o x 1 U(tx)- U(t) _ xO- 1 a(1-C)J ex (t)<

-P-C o~t)-

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(l x-P+ - 1 xT(1+)x -p+ e x

2.= 0

(1- 2 e)(log x)2/2 - 2e log x - c < { U(tx) - U(t) - log x} / al(t) <a0 (t )

(1+e) 2 x (log x)2/2 + 2 e log x + c.

3. y <0

r ________ 1U(tx) - U(t) x - 1(l-e) xT -- - e.xT <'La/t)<

-p+c(l+6) x x -P+C +" e x .

Here a0 and a, are as in Lemma 1.

Proof

For 1) and 3) see Geluk and de Haan 1987, p. 10 and p. 27. For 2) adapt the

proof of Lemma 3 in Dekkers, Einmahl and de Haan (1988).

Lemma 3Let Y - < (nn) be the n-th order statistics from the distribution

function 1 - x (x > 1). If k(n) -* o0 and (2.1) holds with a + sign for y > 0

and a - sign for T < 0, then given e > 0 there exist n0 such that for n > no ,

i - k almost surely (n-o)

1. T 0

( (n-i,n) /Y(n-k,n) - ]

1 a(Y(n-k,n))fY(n-i,n) /Y(n-k,n) [(I-C) -P-C C]

< U~ (n-~n))- U(y (n-kn)) (Y(n-in)./Y (n-kn) )T'

Va1 Y - ') (Y~nia /Y (nk~'1 (1+C) ( n-i,n) /Y(n-k,n) -1+

2. ~y = 0

- 2 e log{Y ( ni,n) /Y (n-k,n)) - CI

{ TI- a~ (n-k,n) - log Y .-~ - log Y n-k n <

1 In-k,n)[(1+e-) 2 Y(n-i,n) /Y(n-k,n)} Cfgy(n-i,n) /Y(n-k,n))l2/

+ 2 elog{Y (n-i,n) /Y (n-k,n)) + eI.

Proof

This is a straightforward application of Lemma 2, taking into account that

y(n-k,n) '0 U*)

Corollary 3

Under the conditions of theorem 1 for each c > 0

4

0<8<1 aO(y.k) r

(n-icc) in probability.

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Proof

The conditions on {k(n)} imply (cf. Dekkers and de Haan 1988, proof th. 2.3)

1im VT(n) a1 (k--) = 0.n-1o

Since I i m Yk (n) = 1 in probability (cf. Smirnov (1949)) andSine Iim (n-k(n))"n-woo

aI e RV0 , also

1 im VY(n) a1 (Y(n-k(n))- 0.n-*wQ

Further by D. Mason ((1982) theorem 3)

sup s { + 2c (1+C) fY(n-[ks],n)/Y(n-k,n) - - 1

0<s<1 (n-[ksjn) Y(n-k n)p+e 1J

is bounded (n-'o) in probability and so is

sup s1+2c[+e) 2 (n2[ks],n)/Y(n-k,n)}Cflog(Y(n_[ks],n))(Y(n-k,n))}2/2 +

O<s<l

+ 2c log{Ym [ks],n/Y(n-k,n)} + C].

The left hand bounds are treated similarly.

Lemma 4

There exists a sequence of Brownian bridges Bn(y) such that for y < 1

s IVv ;(y)1 ([ky],k) (1-y)-- Bk

in probability with h (y) = (1-y)T+l.

Proof

Cor. 3.2.1, p. 27, M. Cs*rg6 (1983).

-9-

Proof of theorem 1

1. Combine Corollary 3 and Lemma 3, further note the distributional equality

f /Y k-1 k-1{Y(n-i,n)/ (n-k,n) k {(k-i,k)i=l.

It remains to show that

x (n-(ks ,n) - X(n-k,n) 1fU(Y(n-fksj,n ) U(Yn-kn)

F n-kn),n (n-k,n) } (n-k,n) n-k,n)

nwith {Y n} as in Lemma 4. For the numerator this is obvious. For thedenominator note that

{i-F(U(y)}/F'(U(y)) = y U'(y).

2. It is well-known (Smirnov 1949) that

R := VT f{Y -i1)n n (n-k,n)

converges in distribution to a standard normal random variable R, say (n-3oo,

k = k(n)/n * 0). Hence

n +Rn/VT U I (n.s.U(Yn-N'n) - U(]) = n +R/ -ds)

nT U- n V U ds -R

in distribution.

3. Follows immediately from Corollary 3 and the reasoning under 2.

Remark

It is possible to formulate the convergence towards the Brownian bridge in one

of the other forms given on p. 26/27 of M. Cs~rgo (1983). This then allows us

to infer the asymptotic normality of Hill's estimate for the inL ,x of regular

variation (Hill (1975)) via an invariance principle.

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Theorem 2

Under the conditions of Theorem 1

sup VT S Y1 X(n [ks ]n) -U(I) +1 5}Y {B (s) +sR}.0O~sl UI nn{in probability (n-.o), where {B) (s)} and R nare as in Theorem 1, R R,

standard normal, and R nand f{B n(s)l are asymptotically independent.

Proof

X n(n-(ksl,n) K~j + }

n U,(n

d U(Y ~ U(n)d{~l (n-[ks],n) iOk i-

V s T+1 EY(n-k,n) U(Y, n U(Y (n-ks],n)) U(Y (n..kn)) +1 -0'-^nU,(n) Y k U(Yk

U'n Y _nk

S(n-k,n) U(Y(n-k, n))

We now investigate the asymptotics of the various terms.

Y YlT U(Y ) ( U)1 .U

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( k ) .k ) (i). 0 (n-wo)

in probability by (2.1), since - Y 1 (n-ww) in probability.E(n-k,n) - nto npoaiiyIn particular

Y ., (Y(n-k,n) (n-k,n))

n un

U, (1)

in probability. Next (note that this term is zero for y = 0)

k Y(nk,n))- - - 1} - TT. R,

n-to (cf. proof of Theorem 1). Finally as in the proof of Theorem 1

U(Y nk,n) - U(]) R~ n U()

(n-poo) in probability.

Remark

The differentiability of F is not required for Theorem 2 and part 1 of Theorem

1: condition (2.5) suffices.

References

M. Cs6rg6 (1983) Quantile processes with statistical applications. Soc.

Industr. and Appl. Math., Philadelphia.

A.L.M. Dekkers and L. de Haan (1988) On the estimation of the extreme-value

index and large quantile estimation. Submitted.

A.L.M. Dekkers, J.H. Einmahl and L. de Haan (1988) A moment estimator for the

index of an extreme-value distribution. Submitted.

t

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L. de Haan and S.I. Resnick (1981) On the observation closest to the origin.

Stochastic Processes and their applications 11, 301-308.

B.M. Hill (1975) A simple general approach to inference about the tail of a

distribution. Ann. Statist. 3, 1163-1174.

D. Mason (1982) Some characterizations of almost sure bounds for weighted

multidimensional empirical distributions and a Glivenko-Cantelli theorem

for sample quantiles. Z. Wahrscheinlichkeitstheorie verw. Geb. 59,

505-513.

S.I. Resnick (1987) Extreme values, regular variation and point processes.

Springer Verlag, New York.

N.V. Smirnov (1949) Limiting distributions for the terms of a variational

series. In Amer. Math. Soc. Transl. Ser. 1, no. 67.

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