Extreme Values in FGM Random Sequences

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1 Journal of Multivariate Analysis 68, (1999) Article ID jmva , available onle at httpwww.idealibrary.com on Extreme Values FGM Random Sequences E. Hashorva and J. Hu sler* University of Bern, Bern, Switzerland Received September 4, 1996; revised January 22, 1998 We consider the multivariate FarlieGumbelMorgenstern class of distributions and discuss their properties with respect to the extreme values. This class was used to consider dependence multivariate distributions and their orderg. We show that the extreme values of these distributions behave as if no dependence would exist between its components Academic Press AMS 1991 subject classifications 60G70, 60E05 Key words and Phrases FarlieGumbelMorgenstern distribution; multivariate extreme values; random sequences. 1. INTRODUCTION In this paper we consider extremes of the FarlieGumbelMorgenstern class of multivariate distributions which was used for the construction of multivariate distributions (see Conway, 1983). It was proposed by Morgenstern (1956), extended by Farlie (1960), and is now known as the FarlieGumbelMorgenstern (FGM) class of distributions. These distributions have a simple natural form with given univariate margals. This class was further generalized to clude distributions with a stronger correlation structure, see, e.g., Johnson and Kotz (1975, 1977) and Cambanis (1977). For a recent discussion of this family of distributions see L (1987), Kotz and Seeger (1993), and Cambanis (1993). Cambanis (1993) considered the question whether this class of multivariate FGM distributions might be the family of fite dimensional distributions of a stationary random sequence or a stochastic process with contuous time. He showed that this class reveals a dependence structure which is rather restricted. It does not clude for stance complete dependence nor strong dependence, general. He noted addition that ``these simple models of dependence may be appropriate for sampled time or spatial processes.'' Our motivation for this short note consists analyzg the extreme value behavior of such stationary or nonstationary random sequences, * Research supported by the Swiss National Science Foundation X Copyright 1999 by Academic Press All rights of reproduction any form reserved. 212

2 EXTREMES IN FGM RANDOM SEQUENCES 213 where we deal with univariate as well as multivariate sequences. Because of the restricted dependence structure the behavior of the extreme values is mostly not fluenced by the dependence structure, asymptotically. A FGM distribution H R n, for n1, is defed with respect to given univariate distributions F i,, by n H(x 1,..., x n )= ` i=1 F i (x i ) {1+ a( j, k) F j (x j ) F k(x k ) = for all vectors x=(x 1,..., x n )#R n where the n(n&1)2 terms a( j, k) are suitable constants, such that H is a distribution function. The univariate margals of H are the given F i. The constants a( j, k) are admissible if the 2 n equalities 1+ = j = k a( j, k)0 hold, for all = j =&M j or 1&m j, where M j and m j are the supremum and the fimum of the set [F j (x), &<x<]"[0, 1]. If F j is absolutely contuous, then M j =1 and m j =0, hence = j =\1. These equalities imply that the coefficients are bounded, for stance by a( j, k) 1[m[M k, M j,(1&m j ), (1&m k )]] 2, which follows immediately usg the bivariate distributions. We assume that the distributions F i are nondegenerated with f j1 M j >0 and sup j1 m j <1. Note that the multivariate distributions are determed by the bivariate margals (by the terms a( j, k) and the univariate F i ) and that their k-dimensional margals are also of the same type. More general FGM distributions were proposed and analyzed the above-mentioned papers. However, we note Section 3 that the behavior of the extreme values of these more general FGM random sequences is not different from the one which is analyzed the followg. A FGM random sequence [X i, i1] is now defed by the univariate margals F i tx i, i1, and a symmetric function a( } ) (that means a( j, k)=a(k, j)) such that the jot distribution of X i1,..., X is given by the FGM distribution n H i1,..., i n (x)= ` h=1 F ih (x i ) {1+ a(i j, i k ) F i j (x j ) F i k (x k ) =,

3 214 HASHORVA AND HU SLER where x=(x 1,..., x n ). The function a( } ) is admissible if for every n1 and [i 1,..., i n ] the equalities 1+ 1j<kn = ij = ik a(i j, i k )0 hold for all = ij. The FGM sequence is stationary iff the univariate margals are all equal, F i (})=F 1 (}), i>1, and the function a( j, k) depends on j, k only through their difference a( j, k)=a( j&k) for all j{k. In this case we use the same notation for the function a( } ) havg one argument only. In the followg the function a( } ) plays the same role the construction of a FGM distribution always even if a( } ) has four arguments. In the same way we can troduce an dependent sequence of random vectors X i, i1, where the distribution H i of X i is a d-dimensional FGM distribution with margals F i, j,, and coefficients a i ( j, k), j, kd. Obviously, if the H i #H 1, the random vectors X i are identically distributed which happens if a i ( j, k)=a 1 ( j, k) and F i, j =F 1, j for all i>1, j, kd. A further extension is possible by defg a multivariate stationary or nonstationary random sequence of d-dimensional random vectors X i. Usg the idea of the construction of the FGM distributions we might defe for stance k H i1,..., i k (x 1,..., x k )= ` h=1 H ih (x h ) {1+ 1h<h$k a(i h, i h$ ) H i h (x h ) H i h$ (x h$ ) = for suitable coefficients a( }, } ). However, this distribution R kd is not of the FGM type. Therefore we defe the FGM random sequence X i, i1, such that for any k1 and i 1 <}}}<i k the kd-dimensional distribution of X i1,1,..., X i1, d,..., X ik, d is a FGM distribution with respect to a function a(}), H i1,..., i k (x 1,..., x k ) k = ` h=1 d ` j=1 F ih, j(x h ) _ {1+ 1hh$k 1l, l$d, (h, l)<(h$, l$) a(i h, i h$ ; l, l$) F i h, l(x hl ) F i h$, l$(x h$l$ ) =

4 EXTREMES IN FGM RANDOM SEQUENCES 215 for admissible coefficients a(h, h$; l, l$), h, h$1, l, l$d, where F i, l (x)= P[X il x]. Note that (h, l)<(h$, l$) means that either h<h$ orh=h$ and l<l$. We consider the partial maxima M n =max X i, n1, the univariate case, and the vector of componentwise partial maxima M n =(M n1,..., M nd )=(max X i1,..., max X id ) the multivariate case. We present the next section some results on the limitg distribution of the maxima for the univariate case and Section 3 for the multivariate case. 2. UNIVARIATE CASE Let [X i, i1] be a sequence of random variables X i where their jot fite-dimensional distributions are FGM. The limitg distribution of the maxima is derived with respect to some suitable normalization u n (x). We want to consider the approximation of P[M n u n (x)]. In this class the dependence between the random variables X i is not very strong. This means that the approximation P[M n u n (x)]r ` P[X i u n (x)]= ` F i (u n (x)) holds. Only a few cases is this approximation not suitable. In the general case of nonidentically distributed X i the followg u.a.n. (uniform asymptotic negligibility) condition is essential for general results. Suppose that for the normalization u n (x) p n, max =sup F i(u n (x)) 0 as n, for the set of x with lim f n > F i (u n (x))>0. For some sequences F i and normalizations u n (x), the limitg distribution G of the maxima M n exists as lim n ` F i (u n (x))=g(x). (1)

5 216 HASHORVA AND HU SLER Note that G (1) is not necessarily an extreme value distribution even under a lear normalization (see Galambos, 1987; Hu sler, 1989a; Falk et al., 1994). In some cases, e.g., nonstationary sequences (see, e.g., Hu sler, 1983) it is reasonable to replace the normalization u n (x) by a boundary [u ni,, n1] which is nonconstant i for fixed n; u ni may depend on x also. The more general u.a.n. condition is now p n, max =sup F i(u ni ) 0, as n. (2) A general theory of extreme values is established for random sequences satisfyg some mixg conditions. Usually a weak distributional mixg condition D=D(u ni ) is supposed with respect to some more general normg or boundary values [u ni,, n1] (see Leadbetter et al., 1983; Falk et al., 1994). For each n and m let n, m be such that for any 1i 1 < i 2 <}}}<i p <j 1 <}}}<j q n with j 1 &i p m, P(X l u nl, l # I _ J)&P(X l u nl, l # I) P(X l u nl, l # J) n, m with I=[i k, kp] and J=[j k, kq]. Condition D is said to hold if n, mn 0, as n for some sequence m n (as n ) with m n p n, max 0. The condition D holds for a FGM sequence if the coefficients a( j, k) satisfy the simple sufficient condition sup a( j, k) 0, n (3) j&k>n We do not know whether this condition holds for all FGM sequences. Cambanis (1993) has shown that the dependence structure of FGM stationary sequences is rather restricted. For stance, (i) m-dependent FGM sequences exist only as dependent sequences (proved for m3); (ii) equal or constant dependence is not possible, aga only dependence is possible; (iii) a positive geometric decay of the coefficients is also not possible. Under condition (3) the limitg distribution of the maxima is derived by the limitg behavior of > F i (u n (x)). Lemma 1. Assume that the FGM sequence X i is u.a.n. and that (3) holds. Then Condition D holds for any boundary values u ni such that lim sup n F i(u ni )<. (4)

6 EXTREMES IN FGM RANDOM SEQUENCES 217 Proof. Note that for any I/[1,..., n], P(X l u nl, l # I)=` F l (u nl ) \1+ l # I l<l$#i a(l, l$) F l(u nl ) F l$(u nl$ ) +. Usg (3) the absolute value of the double sum can be bounded by a(l, l$) F l(u nl ) p n, max + =F l(u nl ) F l$(u nl$ ) l<l$#i, l$&ll 0 l<l$#i, l$&l>l 0 =O \l 0 p n, max ln F l(u nl ) \ ++O = \ F l(u nl ) ln for any =>0 and with suitable l 0 such that a(l, l$) <= for l$&l>l 0. Usg (4) this double sum converges to 0 as n. This holds uniformly for any I/[1,..., n]. Further it implies P(X l u nl, l # I _ J)&P(X l u nl, l # I) P(X l u nl, l # J) ` l # I _ J for any m, uniformly for all I, J. F l (u nl )(1+o(1)&(1+o(1)) 2 )=o(1) K Therefore a general proposition holds for a nonstationary sequence satisfyg (3). Proposition 2. Let [X i, i1] beanu.a.n. FGM random sequence such that (3) and (4) hold with respect to some normalization u n (x). Then as n. If addition (1) holds, then P[M n u n (x)]& ` F i (u n (x))0 (5) P[M n u n (x)] w d G(x) as n. Proof. Note that by the FGM structure of the random sequence we have P[M n u n (x)]= ` F i (u n (x)) {1+ = ` F i (u n (x))[1+o(1)] a( j, k) F j (u n (x)) F k(u n (x)) =

7 218 HASHORVA AND HU SLER iff a( j, k) F j (u n (x)) F k(u n (x))=o(1). (6) Equation (6) is implied by the assumptions as Lemma 1. K The proof shows also the converse statement that (5) implies (6). In the stationary case (6) means that j<kn a( j, k)=o(n 2 ), sce by (4) lim sup n nf (u n (x))<. In the Appendix we show that condition (6) holds always the stationary case. Also it is proved that a slightly stronger condition (6$) implies D for general nonstationary sequences. We believe that (6) implies D most cases; we did not fd a counterexample. Note also that the two conditions are not the same structure and that D implies quite strong extreme value results. 3. MULTIVARIATE CASE Let [X i, i1] be a sequence of dependent random vectors X i R d where their distributions H i are FGM. In the followg the algebraic operations are meant componentwise. Lemma 3. Let [X i, i1] be a sequence of dependent r.v.'s R d with FGM distributions H i. Let [u ni,, n1] be a boundary sequence with u ni =(u ni, 1,..., u ni, d ) such that (4) and the u.a.n. condition hold for each component. Then P[X i u ni, ]& ` ` F i, j (u ni, j ) 0, n. Proof. By the dependence of the X i 's P[X i u ni, ] = ` H i (u ni ) = ` _ ` F i, j (u ni, j ) \1+ a i ( j, k) F i, j(u ni, j ) F i, k(u ni, k ) +&. 1 j<kd Sce p n, max =sup sup F i, j(u ni, j ) 0, the double sum converges to 0 as n and also

8 EXTREMES IN FGM RANDOM SEQUENCES j<kd a i ( j, k) F i, j(u ni, j ) F i, k(u ni, k ) =O \sup \ F i, j(u ni, j ) + p n, max+ =o(1). This implies the statement. K In the case of i.i.d. r.v.'s, X i a limitg distribution of M n exists if every margal F j,, of X i, j belongs to some doma of attraction of an extreme value distribution G j, i.e., there exist normg values a nj (>0) and b nj such that P[M nj a nj x+b nj ]=F n j (a njx+b nj ) w d G j (x) (7) as n. We use the notation a n =(a n1,..., a nd ), b n =(b n1,..., b nd ), and u n (x)=(u n1 (x 1 ),..., u nd (x d )) where u nj (x j )=a nj x j +b nj. Proposition 4. If the i.i.d. FGM random sequence [X i, i1] is such that (7) holds, then P[M n a n x+b n ] w d ` G j (x j ) as n. Proof. Equation (7) implies (4) and the u.a.n. condition for each component. Hence the statement follows by Lemma 3. K Furthermore, we derive easily for i.i.d. sequences P[M n a n x+b n ] d = ` j=1 d = ` j=1 F n j (u nj(x j )) {1+ 1k<ld F n j (u nj(x j ))[1+O(1n 2 )] n a(k, l) F k(u nk (x k )) F l(u nl (x l )) = n w d d ` j=1 G j (x j )=G(x),

9 220 HASHORVA AND HU SLER sce F j (u nj (x j ))=O(1n) and 1k<ld a(k, l)=o(1). This implies the general bound of the convergence rate } P[M nu n (x)]& ` F n (u j n(x j )) } ` F n (u j n(x j )) [1+O(1n 2 )] n &1 =O(1n) uniformly for x. Hence the speed of convergence of P[M n u n (x)] to the limitg distribution depends on the speed of convergence of each component plus this O(1n) term. The dependence of the components M nj is asymptotically vanishg at a rather fast speed. This result can be also extended to the case of dependent but nonidentically distributed random vectors X i. We need to assume that the distribution of the component M nj converges with suitable (lear) normalization u nj (x). Also this multivariate case the limitg distribution is general not an extreme value distribution (see, e.g., Hu sler, 1989a, b; or Falk et al., 1994). Proposition 5. Assume that the i.non-i.d. FGM random sequence [X i, i1] is such that for each component j,, the FGM sequence [X i, j, i1] isau.a.n. and P[M nj u nj (x)] w d G j (x) as n, with suitable normalizations u nj (x). Then P[M n u n (x)] w d ` G j (x j ) as n. The proof follows the same les as the proof of Proposition 2 and Lemma 3. Fally we consider nonstationary multivariate FGM random sequences (troduced Section 1) the followg proposition which is proved along the same les. Instead of (3) we use sup h$&h>m sup a(h, h$; l, l$) 0, n. (8) 1l, l$d Proposition 6. Assume that the FGM random sequence [X i, i1] is such that (8) holds and that for each component j,, the FGM sequence [X i, j, i1] isu.a.n. with respect to the normalization u nj (x). Then P[M nj u nj (x)]& ` ` F i, j (u nj (x j ))0, n.

10 EXTREMES IN FGM RANDOM SEQUENCES 221 If addition (1) holds for each component, i.e., ` F i, j (u nj (x)) w d G j (x) as n, then P[M n u n (x)] w d ` G j (x j ) as n. Remark. Johnson and Kotz (1975, 1977) troduced a more general class of FGM distribution with a stronger dependence between the components. Such a FGM distribution is given by d H(x)= ` j=1 F j (x j ) {1+ d g=2 1 j 1 < j 2 <}}}<j g d g a( j 1, j 2,..., j g ) ` h=1 F j h (x h ) =, where aga the constants a( } ) fulfill some conditions. But the bivariate margals of H are of the same type as the FGM dealt with the begng. Therefore Lemma 3 implies that the bivariate dependence is asymptotically negligible for the events related to extremes. Sce the bivariate dependence of the components of M n implies the multivariate dependence of all components (Hu sler, 1989a; cf. Hu sler, 1994, Theorem 3.4), we get the same results as Proposition 4, 5, and 6 (by assumg obviously stead of (8) a suitably adapted condition) for the more generale FGM distributions. APPENDIX (1) We prove now that a stationary FGM sequence satisfies (6) which means i< j a( j, k)=o(n 2 ) sce by (4) F (u n (x))=o(n &1 ). The constants a( j, k) are admissible if 1+ = j = k a( j, k)0 (9) for any = j =&M j or 1&m j. In the stationary case we have M j =M and m j =m for all j1. In our case of extreme value distributions, we have

11 222 HASHORVA AND HU SLER M=1, which is not used the followg proof. For any choice of ==(= 1,..., = n )#R n, we defe the sets Then (9) can be written as J =[( j, k) = j <0, = k <0], J +& =[( j, k) = j >0, = k <0], J &+ =[( j, k) = j <0, = k >0], J ++ =[( j, k) = j >0, = k >0]. M(1&m) a( j, k)& (1&m) 2 a( j, k) j<k, (j, k)#j &+ _ J +& j<k, (j, k)#j ++ & j<k, (j, k)#j && M 2 a( j, k)1. We sum on all these equations when = # R n such that [j = j <0] =n*. Some of these might be the same, but this does not matter. We get S=S 1 &S 2 &S 3 \ n*+ n. We consider now an element a( j, k) with. This element appears 2 \ n*&1+ n&2 times the first sum S 1,( n&2 )S n* 2 and ( n&2 )S n* 2 &2 3. Hence we have for the sum S on the ( n n* ) choices of = where a( j, k) c \ n*+ n, c =2M(1&m) \ n*&1+ n&2 \ n&2 &(1&m)2 n* + &M \ n*&2+ n&2 2.

12 EXTREMES IN FGM RANDOM SEQUENCES 223 Simplifyg we get c= 2M(1&m)(n&2)! (n&2)! (n*&1)! (n&n*&1)! &(1&m)2 n*! (n&n*&2)! & M 2 (n&2)! (n*&2)! (n&n*)! = \ n*&1+{ _2M(1&m) n*(n&n*)&(1&m) 2 (n&n*&1)(n&n*) n&2 &M 2 n*(n*&1) & = n*(n&n*) = \ n*&1+{ n&2 M 2 n*+(1&m) 2 (n&n*)&((1&m)(n&n*)&mn*) 2 n*(n&n*) =. Choosg n*=w(1&m)(1&m+m) nxt*n, with 0<*<1, the constant c is positive, sce and hence ((1&m)(n&n*)&Mn*) 2 4 c \ n*&1+ n&2 M 2 n*+(1&m) 2 (n&n*)&4 n*(n&n*) = \ n*+ n M 2 n*+(1&m) 2 (n&n*)&4, n(n&1) which is positive for all n large. Thus we get n(n&1) a( j, k) n*m 2 +(1&m) 2 (n&n*)&4 =O(n). j<k Takg all = j =1&m, we get the lower bound from (9), &(1&m) &2 j<k a( j, k). Thus our statement follows and (6) holds. (2) We prove now that the somewhat stronger condition (6$) implies D where a( j, k) F j(u n (x)) F k(u n (x))=o(1) (6$) as n. Let us denote u n =u n (x) for any x fixed. Let A=[X i u n, i # I] and B=[X j u n, j # J] where I and J are disjot subsets of [1,..., n], separated by m used condition D. Let S(I)= j<k # I a( j, k) F j(u n ) F k(u n )=o(1) for any I/[1,..., n]. Then by (4)

13 224 HASHORVA AND HU SLER P(A & B)& ` F } i (u n ) _ J) }S(I i # I _ J P(A) P(B)& ` F } i (u n ) S(J), }S(I)+S(J)+S(I) i # I _ J which are all o(1). Sce this holds for any I, J, it implies D. Note that the separation of I of J is not used which shows the different nature of the conditions. As Lemma 1, we can show that (3) and (4) imply (6$). Note also that D is used to derive stronger results the theory of extremes, as, e.g., the pot process convergence of exceedances to a Poisson process. ACKNOWLEDGMENTS We thank Professor S. Kotz for some comments on an earlier version of this paper and a referee for his valuable comments. REFERENCES 1. S. Cambanis, Some properties and generalizations of multivariate EyraudGumbel Morgenstern distributions, J. Multivariate Anal. 7 (1977), S. Cambanis, On EyraudGumbelMorgenstern random processes, ``Stochastic Inequalities,'' IMS Lecture Notes Monograph Series, Vol. 22, Institute of Mathematical Statistics, Hayward, CA, D. Conway, FarlieGumbelMorgenstern distributions, ``Encyclopedia of Statist. Sciences'' (N. I. Johnson and S. Kotz, Eds.), Vol. 3, pp. 2831, Wiley, New York, M. Falk, J. Hu sler, and R. D. Reiss, ``Laws of Small Numbers Extremes and Rare Events,'' DMV Sem., Vol. 23, Birkha user, BaselBoston, D. J. G. Farlie, The preformance of some correlation coefficients for a general bivariate distribution, Biometrika 47 (1960), J. Galambos, ``The Asymptotic Theory of Extreme Order Statistics,'' 2nd ed., Krieger, Melbourne, FL, J. Hu sler, Limit properties for multivariate extremes sequences of dependent, non-identically distributed random vectors, Stochastic Process Appl. 31 (1989a), J. Hu sler, Limit distributions of multivariate extreme values nonstationary sequences of random vectors, ``Extreme Value Theory'' Lecture Notes Statistics, Vol. 51, (J. Hu sler and R.-D. Reiss, Eds.), pp , Sprger-Verlag, Berl, 1989b. 9. J. Hu sler, Extremes Limit results for univariate and multivariate nonstationary sequences, ``Extreme Value Theory and Applications'' (J. Galambos, J. Lechner, and E. Simiu, Eds.), pp , Kluwer, Dordrecht, N. L. Johnson and S. Kotz, ``Distributions Statistics Contuous Multivariate Distributions,'' Wiley, New York, N. L. Johnson and S. Kotz, On some generalized FarlieGumbelMorgenstern distributions, Comm. Statist. 4 (1975),

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