Conical Extremes of a Multivariate Sample

We introduce multivariate extremes in the direction of a given cone. Convergence results for the number of the kth extremes are obtained for sampling from a distribution having asymptotically independent radial and spherical components and regularly varying tail of the radial component.


Introduction
Let 5(ir.= {Xi,..., Jf"} be a point set of independent identically di.'ttributed t^-ditnensional random vectors sampled from the probability mea.sure fi., and A' be a punctured at the origin cone mK^,d>\.Wc define the kth layer as

^\^.)-{Xr.ti{K^,n 3ifj = t-i} k-\2
where K^ = x^^K is the translated cone with vertex in X E R"*.Intuitively, the Jtth Ij^er is the set of die ith extremes of ^, in the direction /^.The prime examples we have in mind arc {I) the Pareto-optimal points corresponding to the first layer in the direction of the positive orthani, and (2) the total maximum, which may be considered as the first layer in the direction of the COIK, complement to the negati\« orthant.We are interested here in the distributions of random variables counting the number of points in the ^th layer.These distributions depend essentially on both K and ft.
From a more general viewpoint, the first layer can be regarded as the set oimaximal elements[A\ with respect to the binary relation 3i in R** defined as x^y t^ xy ^ K.AIternatively, any scale aixl translation invariant binary relation generates a cone by setting K ~ {x& R**; x'MQ) and the maximal elements are conical extremes.
Two above cases of the counting problem have been considered in the literature under the assumption that fx is either a product of one-dimensional marginal measures or a multivariate normal distribution (2,10,11,12].It is well known, for example, that if /i. is a product measure in R** then the average number of Pareto points is of the order of (log n)'*'', while the probability that the multiple maximum exists is n'""'.
In this paper we focus on a class of distributions ^ already studied in connection with multivariate extreme-value theory [8] and statistics of convex hulls [1,5,6,9].These distributions are characterized by regu lar variation of the tail of the radial component and asymptotical independence of radial and angular components.We show that typically the V,'*''s converge in distribution and the expectations have finite limit.sas ft ^ «^.In the special case of slow variation we calculate explicitly the limiting distributions.

Preliminaries
We define a cone as a punctured at the origin, scaleinvariant Borei set in R'', i.e., 0 ^ K, tK ~ K >;ft>0.Each cone Ls uniquely determined by its spherical base S^-K r\ S , where S denotes the unit sphere.Wc associate with K also the spherical set S_ obtained by reflection abt^ut the origin, 5o -5\ (S+ U 5.) and S:^" S^ H 5-The cone with spherical base CCS will be denoted cone (C).
We fix in what follows a cone K and a multidimen sional probability distribution fi satisfying the following conditions: (i) There exists a > 0 and a probability measure p on 5 such that r>0. ( (ii) For all p-continuous C C 5 (iii) p (im SJ) > 0, and (iv) ft has no atom at the origin.Consider an iid sample from p., 3(f, -{X, X"}, represented in the polar form as the product of radial and spherical components: X, = RiZ,, where /f,-||Xi||, Z, = X;/|[Xj]i.The above conditi<ms on p. have a natural probabilistic interpretation.Condition (i) means that the distribution function of the radial compt>nent, has a regularly varying tail.Condition (ij) is translated as lim P{Z, («,>r} = p(-) and LS to be interpreted as the asymptotic independence of radial and spherical components, where the limiting distribution p does not disappear in the interior of 5* (condition (iii)).The last condition is not essential and assumed for technical reasons.Given a Borel set iJ C R''. we represent the number of the Jtth layer points in fi as the sum of random indicators ft ff.^'\K n fl) = 2 1 (;r,eyf'(ar,.ni.>. and using the iid assumption write for the expectations E#,se*'(^.n B) -nP{X, e ^"XM.)nb}~ "(r;) P{X,Gfi;X, X^^K^,.X,.u..,X,^K^,)= "(j_!) J (t^{K,)t\\-ti(K.)r' d/i(.r) . (3) The following lemmas will be used to estimate these integrals.Lemma 1.There exists T> 0 such that fiiK^) > T(\-n(Bi^))foraIlx&R''.
Proof.Consider first the case where there exists a lirtear isomorphism which maps K onto the positive orthant-Let ;? be the inverse image of the vector (1 1) under this isomorphism.By convexity, K,CK,for3WxGBi.
Ctindition (iii) allows one to select a compact p-continuous set CC S^ with p (C)>0 .It is easy to see that y £ int ^, the sets ^j,, s>0, arc increasing as j i 0 and U,>o K;y = int AT.It follows that C C K" for sufficiently small 5. Furthermore, for small s wc have also A,,c C K^. Indeed, the points of v4].c are rcpresentable as fjc.with f > 1, jr € C, tlius, by convexity, ,t E K^ implies tx G K"y C K,y. Homogeneity implies A,:,,c C K^.It follows now from Eqs.
For arbitrary K one can find a smaller cone K' C K, which is linearly isomorphic to the positive orthant and still has the interior of its spherical base of positive p-measure.This is possible since the spherical <i-simplexcs build a measure-generating class on 5.It remains to note that p.(K^) ^ p.(K't) for any translation, whence the estimate holds in general.□ EV"i> = fl(j_[)J t'-\\-ty^Am{ty A slight modification of the standard Tauberian theorem as found in [14] assures dial the limiting value of this integral for Jt-1 exists iff m(t) is left-differentiable at I-I, in which case the limit and the derivative have the same value.Applying this theorem in the reverse direction one can easily see that all the EV"'*^ 's must have the same limit.D Lemma 3. Assume % is an increasing sequence such that lim nO-F{y"))-*y, y>(i , then with T determined by Lemma 1. Proof.Set

> f (i-T(i-f)r'
where the equivalence can be justified by partial integration, n 3. Pareto-T^ils: a>0 In this section we study the limiting behaviour of V"'*' under the assumption that the regular variation index a in Eq. (I) is positive.Our plan Ls to translate Eqs.(I) and (2) into the convergence, of a suitably normalized sample, to a PoLsson process [6,15] and then apply a continuity argument to prove also the convergence of the V,'*"s.
Compactify R** by adjoining the infinite point ^ and then puncture in the origin.The resulting topological space, say R'', is isomorphic to R"* and canonically embedded into its compactification, bounded from die origin Borel sets B C ft"* being relatively compact.We endow the space A/(H'') of Radon measures with the vague topology: m" A m iff m,(i*)-)m(B) for all relative compacts.

O)
The limiting measure is in MiR"), being infinite on balls centered at the origin as well as on the sets cone (C) with p(C) > 0. In particular, condition (iii) implies V(int K) = ^.Clearly, i'is a pnxluct measure in polar coordinates and has no atoms.
Let ^ be a Poisson point process in R'' widi intensity measure v, arid |, be die random element of Af (R**) associated with the scaled sample a""'^,.Obviously, the operation of taking a layer commutes with rescaling: iP'XaS€") -a^'"{K), a>0.therefore the number of points in each layer remains invariant under scale transformations.One can expect in this situation that V,'" converges in some sense to an analc^ous functional of the Pois.sonprocess.
Define .^•^.Ar-'MPf The resulting integral does not depend on k, as it is suggested by Lemma 2. The integration area can be reduced to 5Mnt S since <^ is infinite on intS..
The following lemma is found in [5].Lemma 4. Let Ebea locally compact.Hausdorff and separable space; ho, hi,... he a uniformly bounded sequence of real measurable functions commonly supported by a relatively compact set; and nu,, m be a sequence of Radon measures on E such that m, -^ mo.The set D-{x & E : 3{x,}, jr.-*jf, h"{x,) -h h{x)} is measurable and if mJD) = 0 then J h"dm" -*/ h(dmn.Now we are ready to prove a convergence result.Theorem L As.sume (i)-{iv).a> 0, and v{-aK)-0, ( Then for all k= 1.2 Proof.By SkortAod's theorem we can find random point measures |,,fGAf(ft'') satisfying ^"=^", i = f and i"-^ ^ a.s.Thus to prove the convergence in distributicm (10) it suffices to consider the case ^, -^ | a.s.In what follows wc fix a typical realization of i and assume n sufficiently large.
Since i'(ini K) = ^.i lays in the cone interior infinitely many points.Select *; of thent, say jr, xt.Pick r sufficiently small to satisfy B, C fl*., Kx, as well as f((?B,) = 0 and also i*.n {.t, x,} = 0.The complement B", is relatively compact hence the processes ^ and £. have there a finite number of points.points may be labeled so thaty,j -»y,, as it follows from the vague convergence.By the construction, any translated cone /Tj, with x EL Br contains at least k points of ^, thus B, n ^Xi) = 0 and also «, 0 ,T'\^,) = 0 for J=-l k.For y, e -inl K, the cone K^^ contains a vicinity of the origin, where £, has infinitely many points.Therefore y, € n)., ^'ia y"j e n*., ^«(4).
The condition shown in Eq. ( 9) assures that no one of yi yp lies on -t9K, almost surely.For yi ^ -cl K, the shifted cone AT,, is bounded from the origin.Therefore there exists an open vicinity of clfUf= 1 Ky) which is still relatively compact, and hence contains at most a finite number of points in addition to y, J-p • By Eq. ( 10).({dKy,) = 0 a.s.Again the poinwise convergence implies ^ (^v"^ = f(Ar,,), whence (V.'" V/") -(V" V"0°and thus(M).Now turn to the convergence in mean.It Ls enough to prove Eq. ( 12) for the first layer, A -1.It Ls easy to see that Take a point x ^-c\ K and a sequence x" -» x and consider the indicator functions of the sets K^ and K,^ as the h's in I^mma 4. The divergence set D is (?/f" whence by (9) and the lemma v" {K^J -> v(K,).
The right-hand side here tends to zero as r-^.
Putting this all together and comparing with Eq. ( 8) wc conclude lim sup EV'i' « EV".

IimiiifEV,"'>EV'
n->=c follows from the convergence in distribution D Remark.The continuity conditions of Eqs. ( 9) and (10) are actually some properties of the spherical measure p.The first one trivially translates as p {-fiK) = 0, but we have not been able to find a re-formulation for the second.Sufficient conditions for Eq. ( 10) are: p is non-singular, and ^S» lies in a (n-2>-dimcnsional set; or K is convex, ^K has no two-dimensional facets aixl p(aS.)= 0.

n-i^ 4
The first layer is either empty or just one point, maximizing both components, thus this mean value coincides with the limiting probability of the total maximum.The limiting distribution and higher moments of the V^*' 's can be, in principle, expressed in terms of some integrals similar to Eq. ( 8).These expressions do not seem tractable by analytical methods because of the complicated integration domains.

Slowly Varying Tails: of = 0
The case of slowly varying radial tail, with a = 0 in Eq. ( 1), is of special interest.The above Poisson approximation method does not work, since the sample cannot be rescaled to provide a non-degenerate limit.To get around, we extend here a method already exploited in [IJ, where the number of convex hull extremes of a sample under slighly stronger as.sumptions on the dLstribution has been studied.
We assume for technical reasons that F is continutxis though, in fact, slow variation is all that is needed.
Mailer and Resnick [13] proved that slow variation is equivalent to For it = I we have the limiting mean of the number of Parcto points.Now suppose K is the complement to the negative quadrant.The Jtth layer consists of those X,'s which exceed all except some it-1 sample points in both components.We get Our convergence results effectively exploit this fact combined with the asymptotic independence of the concomitants shown next.
Lemma 5. Assume that F(r) = pi(Br) is continuous and Eq. ( 2 (for large r uniformly in n) d^oCr*) + f ^ P(C,)... p{Ct) + €, as «->« , where we have used Eq. ( 14) and applied an argument similar to that in Lemma 3. Asymptotically, the probability is factorized, whence the statement D To prove the convergence we combine in what follows the above lemma and Eq. ( 13).The idea is that the points with top layer ranks have also small ranks in the radial components.On the other hand, conical extrcmality of the points with small layer ranks is determined by their, almost independent, spherical components.
Remark.Given a binary relation, say 'M, on a sampling space, and a random sample X, tlicrc are two natural ways to define the "/tth extremes" o/9if,; (1) sample elements X, which are in vfi with all other sample elements with the exception of some k-\ points; or (2) the elements X, such that there are exactly k-\ sample points which are in the relation with X,.Tn the theory of partially ordered sets extremes {k = IJ of the first type are called the greatest p<^ints, of the second type-maximal [41.This is best illustrated by the natural partial order of R'': total maximum is the greatest point, while Parcto set consists of maximal points.If the binary relation £S is generated by a cone K, as mentioned in Introduction, then the X'-extremes are maximal points, while the -/L ' -extremes are the greatest points w.r.t.'M.Baryshnikov [3] has proved that the asymptotic upper btnind for (he pr<xJuct of expectations of the numbers of the extremes of both types is at most I, for any fixed 9? and k.Theorem 4 shows that this bound is sharp.
Remark.Normal multivariate distributions can be viewed as the case of fast decreasing radial tails, ci = ^.The mean number of conical extremes demonstrates typically the following behavior: for any A, EV' infinitely grow.sif K is contained in a half-space, and lends to zero if K contains a half-space [10,11],

Volume 99 ,
Number 4, July-Augus> 1994 Journal of Research of the National Institute of Standards and Technology