A N O T E O N T I I E COVEKGENCE O F BIVAKIATE EXTREME ORDEK STATISTICS
Ilaroon Mohamed Barakat
Let Xi = (Xlj, X,$, j = 1, 2,
...,
n be a random sample from a bivariate population X = (X,, X,) with d.f. F(x) = F(x,, x 2 ) Let denote the j-th order statistic of theXi
sample values, j = I , 2,...,
n; i = 1, 2. For a, b, X, y EK*,
write ax+
b and-X
toJ denote respectively the vectors (alx,
+
b,, a2x2+
bJ and(5
,%)
and X 2 y toYl
Y2mean xi I
yi,
i = l , 2. Write Zk,i':n, Wk,k,:n and Vd,k.n to denote respectively therandom - l: + l:,, - + L:n), (Xl,k::n, and (X1,k:71, X2,n - + l : n ) , where k and k' are any positive integers. Let further G(x) = P(X > X) and
H,,,,:,(x) = P(Z,,,:, I X). Finally, for the i-th (i = 1, 2) component of the vector X, let $(xi) and Gi(xi) be the marginals of F(x) and G(x) respectively, while E3,,,:,(xl) and H k,,,(xz) be the marginals of Hk,k':n(x). It can be shown that (see Baralrat, 1990, 1999)
and
we have
0 < 7, v 7, 5 lim inf n(1 - F(u, )) 5 lim sup n(1- F(u,)) 5
a+- n-+-
Let us now prove the relation
which is equivalent to
. P(x)
+
0 , as X+
x",
(2.10)1 - F(x-)
where P(x) = F(x) - F(x -). O n the other hand (2.7) is equivalent to
where P,(x,) = F,(x,) - F,(x, -), z = 1, 2 and P(x) = F(x) - F(x -). Suppose, now that (2.10) does not hold. Then, there exist E > 0 and a sequence {V,) = {(V,,, v2,2)) of vectors such that V,
+
X' andO n the other hand, for sufficiently large n, by (2.11) we have, Pi(v,,,) < EG~(v,,,
-1
,
i = 1 2 :which yields
Now, by using the obvious relation Pl(xl)
+
P2(xZ) - P(x) = G(x - ) - G(x) 2 0 , we getTherefore, by (2.12), (2.14) and by using the relation
l ( q
+
5)
2,
GF2 2 F. 2we get
for all sufficiently large n , which contradicts (2.13), i.e. (2.9) is proved.
A note on the coverfence of liivaviate extreme order statistics 3 1
(2.8) clearly guarantees the existence of a real number 0 < T < W , 0 < zl v t2
<
T T,+
z2 < W , such that n(1 - F(u,)) -+ z, which, in view of the relation 1 - F(u,) = G,(u,,)+
G2(u2J - G(u,), leads immediately to (2.3) (since clearly U,+
X'). O nthe other hand, if (2.15) does not hold, then
0 < lim inf n(l - F(u, )) < lim sup n(l - F(un-)) < W :
n-+W l/ -+
by which we can select a subsequence nk such that 1 - F(u,
1
lim
~ . , ~ l - F ( u fq, -) + 1 , which contradicts (2.9).
Remark 2.1. Although the statement of Theorem 2.1 will not be changed if all d.f.'s (F(x), Fl(xl) and F2(x2) are assumed to be left continuous, the relation (2.7)
)
should become
-+
1, as xi -+ X:, i = l , 2. Similar obvious changes will be G i b i + )done for (2.9), (2. l!)), etc.. .
W e now turn to the proof of Theorem 2.1.
Proof of Theowm 2.1. Assume that the marginals Hk &l,) and H ,kl n(u2n) con-
verge to a nondegenerate limits (i.e., positive and finit; numbers). Then, in view of Lemma 2.1, (2.1) and (2.2) are satisfied. Therefore, in view of Lemma 2.2, (2.7) is also satisfied and the first part of the theorem is now followed by applying the sec- ond part of Lemma 2.2. To prove the second part it suffices to use Geffroy condi- tion (see Geffroy, 1958) which is a sufficient and necessary condition for the inde- pendence of the limit marginals (see Barakat, 1998, 2000). However, in view of the relation G(x) = Gl(xl)
+
G2(x2) - (1 - Fix)), Geffroy condition is equivalent toNow the second part of theorem follows immediately by using (2.16) and the obvi- ous relation
Dqartment of Mathematzcs, Fucdty of Sczence Zagazzg Un2uersztyl, Zagazzg, bgypt
ACKNOWLEDGEMENT
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RIASSUNTO
I n questa nota si tnohtra chc per ogni vettote b i ~ a r i d t o di statistiche cl'ordine estreme, cristc almcno un,l succcssione di Lcttori reali per cui la f ~ r n ~ i o r i e di distriluzione (cl f ) con- verge ad un liiuite h i t o se e solo re lc marginali hanno limiti tiniti Inoltre tale litnite si pub s c ~ i v e r c come il prodotto delle nwginali se la d I h i v a r i ~ t a d~ cui sono calcolatc Ie stdtisti-
che d'ordine i: quella di u m variahile casuale dlpendcntc negattve quadmnte
SUMMARY
A note on the convergence of bivariatt~ extreme order stutistics
I n this note an interesting fact is proved that, for any vector of bivariate extreme order statistics there exists (at least) a sequence of vectors of real numbers for which the distrihu- tion function (d.f.) of this vector converges to a nondegenerate limit if and only if its marginals converge to nondegenerate limits. Moreover, the limit splits into the product of the limit: rnarginals if the bivariate d . i . , irom which the order statistics are drawn, is of negative q ~ ~ a d r a n r dependent random variables (r.v., S ) .
