An Alternative Characterization of Hidden Regular Variation in Joint Tail Modeling

Transcription

1 An Alternatie Characterization of Hidden Regular Variation in Joint Tail Modeling Grant B. Weller Daniel S. Cooley Department of Statistics, Colorado State Uniersity, Fort Collins, CO USA May 22, 212 Abstract In modeling the joint upper tail of a multiariate distribution, a fundamental deficiency of classical extreme alue theory is the inability to distinguish between asymptotic independence and exact independence. In this work, we examine multiariate threshold modeling based on the framework of regular ariation on cones. Tail dependence is described by an angular measure, which in some cases is degenerate on joint tail regions despite strong sub-asymptotic dependence in such regions. The canonical example is a biariate Gaussian distribution with any correlation less than one. Hidden regular ariation (Resnick, 22), a second-order tail decay on these regions, offers a refinement of the classical theory. Preious characterizations of random ectors with hidden regular ariation are not well-suited for joint tail estimation in finite samples, and estimation approaches thus far hae been unable to model both the heaier-tailed regular ariation and the hidden regular ariation simultaneously. We propose to represent a random ector with hidden regular ariation as the sum of independent first- and second-order regular arying pieces. We show our model is asymptotically alid ia the concept of multiariate tail equialence, and illustrate simulation methods with the biariate Gaussian example. Finally, we outline a framework for estimation from our model ia the EM algorithm. 1 Introduction Classical multiariate extreme alue theory proides a theoretical framework for describing the joint upper tail of a random ector. Modeling approaches based on classical theory are based on the limiting distribution of componentwise maxima. An extension to multiariate threshold exceedances is based on the framework of regular ariation. This approach describes the limiting joint tail of a random ector as the product of a radial component which decays like a power function and an angular component goerned by a limiting angular measure on the unit sphere under a chosen norm. Oer the past 15 years, it has been recognized that such an approach can fail in applied modeling of joint tails. The fundamental shortcoming is that the first-order angular measure is degenerate on some joint tail regions, thus masking possible (and potentially strong) dependence structure at sub-asymptotic leels. Ledford and Tawn (1996) proided a first attempt at accounting for this sub-asymptotic dependence, using the example of the biariate Gaussian distribution with correlation ρ < 1. Following Ledford and Tawn (1996), many papers hae offered refinements to the classical theory in attempts to resole the flaw of the first-order limit. Ledford and Tawn (1997) and Ramos and Ledford (29) focus specifically on modeling biariate joint tails in the case that the first-order limit fails to capture dependence. Heffernan and Tawn (24) offer a conditional approach, while Coles et al. (1999) examine measures of dependence in the asymptotic independence setting. Draisma et al. (24) and Peng (1999) offer other approaches to joint tail estimation. From a probabilistic perspectie, the concept of hidden regular ariation (Resnick, 22) offers a mathematical structure for describing sub-asymptotic dependence, and is based on a generalization of the methods of Ledford and Tawn (1996, 1997). Hidden regular ariation is essentially a second-order regular ariation on regions where the first-order limit is degenerate. More treatment is gien in Maulik and Resnick (24), Heffernan and Resnick (27), and Mitra and Resnick (21). More recently, De Haan and Zhou (211) offered a refinement on Ramos and Ledford (29), offering an alternatie polar coordinate transformation for modeling joint tails. From a modeling standpoint, the joint tail approach of Ledford and Tawn (1997) and Ramos and Ledford (29) fails to simultaneously account for the first-order limiting tail structure, and it is not immediately clear how to extend such methods into dimension greater than two. Maulik and Resnick (24) offer a 1

2 representation of hidden regular ariation as a mixture of a first-order and second-order component. While this proides an asymptotically alid characterization, a mixture representation is clumsy for finite samples and is difficult to justify intuitiely. In this work, we offer a characterization of a random ector with hidden regular ariation as the sum of independent first- and second-order pieces. An alternatie to Maulik and Resnick (24), our characterization is more amenable to finite-sample representation and estimation. This representation is asymptotically justified ia the concept of multiariate tail equialence (Maulik and Resnick, 24). When the hidden measure is finite, we can simulate realizations from our model; when the measure is infinite, we offer a slight adjustment to our simulation methods. A framework is outlined for estimation from the model ia the EM algorithm. We first reiew the concepts of multiariate regular ariation, hidden regular ariation, and tail equialence. To describe tail dependence in practice, one typically transforms each marginal distribution to a common, heay-tailed marginal; often, the transformation is to unit Fréchet: F Z (z) = exp{ z 1 } (Ledford and Tawn, 1997; Ramos and Ledford, 29). The Fréchet marginal case is a special case of multiariate regular ariation, which describes the joint tail as decaying like a power function. A decomposition into polar coordinates arises, and tail dependence can be characterized by a limiting angular measure. Hidden regular ariation offers a second-order analogue of multiariate regular ariation. A polar coordinate decomposition also arises; howeer, the resulting limiting angular measure is not guaranteed to be finite. Finally, it is through the concept of tail equialence that we show the asymptotic alidity of our model. 1.1 Multiariate Regular Variation Multiariate regular ariation on cones proides a probabilistic framework for describing tail dependence and modeling multiariate threshold exceedances. C is a cone of R d if for a set A C, ta C for any t >. We assume some familiarity with regular ariation of functions in the uniariate case; the interested reader is referred to Bingham et al. (1989) and de Haan (197). Let M + (C) be the space of Radon measures on C. Following Resnick (27), we say that a random ector Z with alues in a subset of [, ) d and common marginal distributions is regular arying on C = [, \{} with finite limiting measure ν if there exists a function b(t) as t such that on C, [ Z b(t) ν( ), (1) in M + (C) as t and denotes ague conergence of measures (Resnick, 27). It follows that there exists α such that the limiting measure ν in (1) has the scaling property ν(ca) = c α ν(a), c > (2) for any relatiely compact set A C, where α > is called the tail index. The joint tail power-function behaior can be seen in (2). The function b(t) is regular arying of order 1/α, which following Resnick (22) we denote b(t) RV 1/α. The homogeneity property (2) suggests a transformation to polar coordinates. Let be any norm on C, and consider the unit sphere N = {z C : z = 1}. Define the bijectie transformation T : R d [, ) N ia T (Z) = ( Z, Z Z 1 ). The measure ν can then be expressed in terms of the new coordinate system (r, θ) ia ν = ν α H (3) where ν α is a Pareto measure; i.e. ν α ((x, ) = cx α, c >, and H is a non-negatie measure on N satisfying the balance condition θ 1 H(dθ) = θ j H(dθ), j = 2,..., d. (4) N N H is called the spectral measure or angular measure. When α = 1, a common choice for is the L 1 norm, in which case H is a measure on the unit simplex N = d 1 = {z C : z z d = 1} (Coles and Tawn, 1991; Ballani and Schlather, 211; Cooley et al., 21). The polar coordinate representation (3) of the measure ν can be seen by considering, for any r > and 2

3 Borel set θ Θ, ν({z E : z > r, z 1 z θ}) = r α ν({r 1 z : z > r, z 1 z θ}) = r α ν({r 1 z : r 1 z > 1, r 1 z 1 (r 1 z) θ}) = r α ν({y E : y > 1, y 1 y θ}) = r α H(θ), where y = r 1 z. Thus with respect to the coordinate system ( y, y y 1 ), we hae that ν is a product measure. Finally, we note that by appropriate choice of normalizing function b(t), H( ) can be made to be a probability measure. 1.2 Hidden Regular Variation and Tail Equialence It is possible that the limiting measure ν in (1) places zero mass on pie-shaped regions {z C : z z 1 N} of the cone C. In such cases, the normalizing function b(t) obliterates any finer structure of the random ariable on such regions, if such a finer structure exists. The corresponding angular measure H thus places zero mass on corresponding regions of the unit sphere N. A classic example is the joint upper tail of a multiariate normal random ariable with correlations less than one (Ledford and Tawn, 1996). This prompted Resnick (22) to formulate the concept of hidden regular ariation. Consider a subcone C C with ν(c ) =. A random ector Z is said to possess hidden regular ariation if, in addition to (1), there exists a non-decreasing function b (t) with b(t)/b (t) such that [ Z b (t) ν ( ) (5) as t in M + (C ). The measure ν decomposes into a product of Pareto measure ν α and positie Radon measure H on N = N C. The function b (t) RV 1/α, with α > α; thus, Z has a lighter tail on C than on C. As N may not be a relatiely compact set of N, H may be either finite or infinite; see (Resnick, 22; Maulik and Resnick, 24; De Haan and Zhou, 211) for details. Finally, ν is homogeneous with tail index α, analogous to (2). The concept of multiariate tail equialence was introduced in Maulik and Resnick (24). Consider random ectors Y and Z taking alues in [, ) with distribution functions F and G, respectiely. Y and Z are said to be tail equialent on the cone C C = [, \ {} if there exists a scaling function b (t) such that [ [ Y b (t) Z ν ( ) and b (t) cν ( ) (6) in M + (C ) for some constant c (, ) and measure ν on C. The definition (6) implies that the extremes of samples from Y and Z hae the same asymptotic properties on C, up to a scaling constant. Following Maulik and Resnick we write Y te(c ) Z. The remainder of the paper is structured as follows: in Section 2 we describe the construction of our sum representation and show that our representation is tail equialent to a random ector with hidden regular ariation. Section 3 demonstrates simulation from our representation for a random ector with Gaussian dependence structure. We conclude in Section 4 with a discussion and framework for estimation of our model. 2 Regular Varying Sum Representation To represent random ectors possessing hidden regular ariation, Maulik and Resnick (24) consider mixtures of random ectors with differing tail indices and angular measures. Maulik and Resnick show such mixtures are tail equialent to a random ector possessing hidden regular ariation. In contrast to Maulik and Resnick, here we consider sums of such random ectors. We show that such sums can be constructed to be tail equialent on both C and C to a random ector with hidden regular ariation. 3

4 Consider a random ector Z [, ) d which is multiariate regular arying on C with limit measure ν; that is for some function b(t), [ Z b(t) ν( ) (7) as t in M + (C). Without loss of generality, assume that the resulting angular measure H is a probability measure. Further assume that Z exhibits hidden regular ariation on a subcone C C. That is, ν(c ) = and there exists a function b (t) with b(t)/b (t) such that [ Z b (t) ν ( ) (8) as t in M + (C ). We construct a sum of regular arying random ectors that is tail equialent to Z on both the full cone C and the subcone C. Define a random ector Y = RW taking alues in [, ) d, where P(R > r) 1/b (r) as r and W H( ), where H and b(t) are as aboe. Recall that ν(c ) = and thus H(N C ) =. We thus define W to be such that P[W A = if H(A) =. Assume that the quantities R and W are independent. It follows that, on C [ Y b(t) ν( ) (9) as t (Maulik and Resnick, 24). Now consider a random ector E (, d defined on the same probability space and independent of R and W which is multiariate regular arying on C with tail index α > α. Assume and P( E > r) 1/b (r) as r, [ E b (t) ν, (1) in M + (C ), where ν is as aboe. Further assume that E is not regular arying of any index α α on C \ C. To reiew, we construct Y to be regular arying with tail index α with support on C \ C ; Y has no hidden regular ariation on C. E is regular arying on the subcone C with tail index α > α and limit measure ν ; that is, E has the same tail behaior as Z on C. It can be shown that mixtures of Y and E are tail equialent to Z on both C and C ; see (Resnick, 22; Maulik and Resnick, 24; Mitra and Resnick, 21). In many applications, it may be more natural to represent Z as a sum of the random ectors Y and E. Next we show Z te(c) Y + E and (11) Z te(c) Y + E. (12) The result (11) follows from Jessen and Mikosch (26); we reiew the proof below. Following a similar argument, we proe (12). 2.1 Tail Equialence on C With Y and E defined aboe, we adapt Lemma 3.12 of Jessen and Mikosch (26) to show tail equialence on the full cone C. Consider a relatiely compact rectangle A C; that is, A is bounded away from. This class of sets A generates ague conergence in C; thus it is sufficient to show lim b(t) A = lim [ Z b(t) A = ν(a). (13) 4

5 Assume without loss of generality that A = [a, b = {x C : a x b}. For small ɛ >, define a ɛ = (a ɛ 1,..., a ɛ d ), and define b ɛ analogously. Define the rectangles A ɛ = [a ɛ, b and A ɛ = [a, b ɛ. For small ɛ, the rectangles A ɛ and A ɛ are relatiely compact in E, and A ɛ A A ɛ. Finally we note that ν( A) = ; there is no mass on the edges of A. For small ɛ > and fixed t >, P A b(t) Thus = P b(t) A, E b(t) > ɛ [ Y + E lim sup A b(t) For the lower bound, recognize [ Y + E P A b(t) and so [ Y + E lim inf A b(t) + P A, E [ Y b(t) b(t) ɛ P [ E > b(t)ɛ + P b(t) A ɛ. lim sup = lim sup [ E > b(t)ɛ + lim sup t 1 α/α ɛ α + lim sup = ν(a ɛ ) ν(a) as ɛ. [ Y b(t) A ɛ [ Y b(t) A ɛ [ Y P b(t) Aɛ, E [ Y b(t) ɛ P b(t) Aɛ P [ E > b(t)ɛ, [ Y lim inf b(t) Aɛ = ν(a ɛ ) ν(a) as ɛ. lim inf [ E > b(t)ɛ Collecting the upper and lower bounds, and using the fact that A is a ν-continuity set, we achiee the desired result [ Y + E ν( ) b(t) in M + (C). 2.2 Tail Equialence on C For (12) to hold, it is sufficient to show the following result: Theorem. For Y, E, b (t), and ν defined as aboe, [ Y + E ν ( ) (14) b (t) as t in M + (C ). Proof. As aboe, it suffices to consider any rectangle A which is relatiely compact in C, and show that [ Y + E lim A = ν (A ). b (t) Without loss of generality assume A = [c, d = {x C : c x d}. For small ɛ >, define the rectangles A ɛ = [c ɛ, d and A ɛ = [c, d ɛ, with c ɛ and d ɛ defined analogously to a ɛ and b ɛ aboe. For small ɛ, A ɛ and A ɛ are relatiely compact in C, A ɛ A A ɛ, and ν ( A ) =. Recognize that for small ɛ > and fixed t >, [ [ E E P b (t) Aɛ = P b (t) Aɛ, Y [ E b (t) ɛ + P b (t) Aɛ, Y b (t) > ɛ [ [ Y + E E Y P A + P c, b (t) b (t) b (t) > ɛ. 5

6 Thus by definition of Y and E and independence, lim inf b (t) A lim inf [ E b (t) Aɛ = ν (A ɛ ) lim inf t(t 1 c α )(t α/α ɛ α ) = ν (A ɛ ) ν (A ) as ɛ, since A is a ν -continuity set. For the upper bound, we employ the fact that H(C ) =. For fixed t, [ Y + E P A = P A, Y b (t) b (t) b (t) ɛ + P b (t) Notice that I is bounded aboe by = I + II [ E P b (t) A ɛ. Recalling that by construction P[Y/b (t) A ɛ =, II = P A, Y b (t) b (t) > ɛ, E b (t) A, + P A, Y b (t) b (t) > ɛ, E b (t) / A, [ [ E Y d P c, b (t) b (t) > ɛ + P i=1 E i b (t) Then lim sup b (t) [ E A lim sup b (t) A ɛ = ν (A ɛ ) + lim sup = ν (A ɛ ) ν (A ) as ɛ, [ [ E Y lim inf b (t) c P b (t) > ɛ Y b (t) Y b (t) [ E + lim sup + lim sup A, Y b (t) > ɛ / A ɛ / A ɛ > ɛ, Y b (t) > ɛ. b (t) c [ d i=1 E i > ɛ P b (t) [ Y P b (t) > ɛ [ Y b (t) > ɛ t(t 1 c α )(t α/α ɛ α ) + lim sup t(t 1 ɛ α )(t α/α ɛ α ) by independence and ν -continuity of A. Finally, putting together the upper and lower bounds yields the desired result (14). Remark 1. Heuristically, the scaled random ector (Y+E)/b (t) can only land in C when Y is small and E is large. Suitably normalized large alues of Y will conerge to points outside of C, and by independence, the probability of Y and E being simultaneously large is asymptotically negligible. Remark 2. The proof relies on Y being constructed in such a way that P[Y C =. Such a condition gies conergence to the measure ν on C. The result may not hold in general if Y has angular measure H only in the limit and exhibits hidden regular ariation on C. We do not impose such additional conditions on the construction of E. Remark 3. Because the measure H can be made to be a probability measure, simulation of realizations of the random ector Y is often quite tractable, especially in low dimensions. The angular measure H of E may be infinite on C, making simulation more difficult. In some cases, H can be made to be a probability measure under an alternatie transformation (Mitra and Resnick, 21; De Haan and Zhou, 211). This still may pose difficulty in simulation; see Section 3 for an example. 6

7 3 Example: Biariate Gaussian Dependence We now demonstrate an example of Z for which we can simulate a tail equialent representation Y + E. We explore the case of asymptotic independence plus hidden regular ariation in dimension d = 2, with the biariate normal distribution as the classical example (Ledford and Tawn, 1996). Through simulation, we see that the sum representation results in a more realistic representation of the random ector Z. Here the resulting hidden measure is not finite, and difficulty arises near the boundaries of the subcone. We reiew preious attempts to address this difficulty, and propose a noel method which accommodates our sum representation. Up to this point, we hae considered polar coordinate transformations defined by any norm on C. Unless noted otherwise, in this section we consider the L 1 norm transformation defined by r = z 1 + z 2 and w = z 1 /r. This transformation is common in most multiariate extreme alue analyses with Fréchet marginal ariables. Consider the biariate random ector Z = (Z 1, Z 2 ), where Z i = 1/ log Φ(X i ), i = 1, 2, and (X 1, X 2 ) follow a biariate Gaussian distribution with correlation ρ < 1. Here Φ( ) is the standard Gaussian distribution function. Sibuya (196) showed that asymptotic independence holds, i.e., we can find b(t) RV 1 such that [ Z b(t) ν = ν 1 H, (15) in M + (C), where H consists of point masses at the axes N {x C : x 1 x 2 = }. If b(t) = 2t, H is a probability measure with point masses of 1/2 at w = and w = 1. An exploration of the second-order regular ariation of Z was proided by Ledford and Tawn (1996, 1997). Ledford and Tawn formulate this in terms of the joint surior function F (z 1, z 2 ) := P[Z 1 > z 1, Z 2 > z 2. Ledford and Tawn (1997) show F (z 1, z 2 ) (z 1 z 2 ) 1/(1+ρ) L(z 1, z 2 ; ρ)(1 + O[1/ log{min(z 1, z 2 )}), (16) where L(z 1, z 2 ) is a slowly arying function gien by (Ledford and Tawn, 1996) L(t, t; ρ) = (1 + ρ) 3/2 (1 ρ) 1/2 (4π log t) ρ/(1+ρ). (17) The random ector Z also exhibits hidden regular ariation. Consider a set [z, for z = (z 1, z 2 ) with z 1, z 2 >. One can show [ Z [z, (z 1 z 2 ) 1/2η =: ν ([z, ) (18) b (t) as t, where the function b (t) := 2U (t), with U(t) = (2t)1/η L(2t, 2t), (19) for L gien by (17). Ledford and Tawn refer to η = (1 + ρ)/2 (, 1 as the coefficient of tail dependence. It is easily shown for sets of the form A(r, B) = {z C : z > r, z z 1 B} that ν (A(r, B)) = r 1/η H (B), where B is a Borel set of N = (, 1). The polar coordinate density of the measure ν is gien by ν (dr dw) = η 1 r 1 1/η H (dw), where H (dw) = 1 4η {w(1 w)} 1 1/2η ; (2) see, for example, Beirlant et al., 24, chapter 9. Note that H (N ) = (,1) H (dw) = +, thus the hidden measure ν is infinite on C. The fact that the hidden angular measure is infinite poses difficulty in finite-sample simulation of the joint tail of Z. Because the hidden measure dierges near the endpoints of N, one always encounters 7

8 difficulty near the axes of C. Seeral authors hae explored ways to remedy this problem. Mitra and Resnick (21) propose an alternatie transformation T (z 1, z 2 ) = (z (2), z/z (2) ), where z (2) = min(z 1, z 2 ). The limit measure ν can then be decomposed into the product of a Pareto measure and probability measure on Ñ = {z C : z (2) = 1}. Howeer, this approach aoids behaior near the axes, and it is not clear how one would simulate random ectors which are tail equialent to Z under this alternatie representation. More recently, De Haan and Zhou (211) offered an alternatie for characterizing the random ector Z. De Haan and Zhou cleerly define a transformation on C ia T (z 1, z 2 ) = (s, ), where s = (z z 1 2 ) 1 and = s/z 1. They then show that the limiting measure of the joint tail of the normalized ector Z 1/η can be decomposed into the product of a Pareto measure and a finite measure H on N = {z C : (z z 1 2 ) 1 = 1}. Specifically for the Gaussian dependence example, H is proportional to a Beta distribution with parameters (1/2, 1/2). While De Haan and Zhou (211) offer a method for constructing a random ector which is tail equialent to Z, their simulation method still encounters problems near the axes of C, which we illustrate in Section Simulation from Sum Representation Because the hidden measure with density H (dw) gien by (2) is infinite on (, 1), one cannot simulate from it directly. As an alternatie, we propose an approximation to H (dw) by restricting the subcone C to C ɛ = {z C : z 1 z 1 N}, ɛ where N ɛ = [ɛ, 1 ɛ for some ɛ (, 1/2). The density (2) can then be made to be a probability density on N ɛ ia H(dw) ɛ = H (dw)/h (N) ɛ for w N. ɛ One can then simulate realizations from H ɛ ia an accept-reject algorithm or other sampling method. We proceed to simulate realizations of Ẑ = Y + Ê which is tail equialent to Z on C and Cɛ. Define Y as follows: let R follow a Pareto distribution with P[R > r = 2/r for r 2. Draw a Bernoulli(.5) random ariable B independently of R, and let Y 1 = RB, Y 2 = R(1 B). For a fixed sample size n, draw R independently of Y = (Y 1, Y 2 ) T with R such that { d ɛ,n x 1/η if x > (d ɛ,n ) η P[R > x = 1 otherwise, { where d ɛ,n = (2U (n)) 1/η H (N) ɛ }. n Draw n independent realizations of W from the density H ɛ (dw) independently of Y and R. Define Ê = (Ê1, Ê2) ia Ê 1 = R W and Ê2 = R (1 W ). Then for any set A(r, B) = {z C ɛ : z > r, z 1 z 1 B} with B a Borel set of N, ɛ [ [ Ê np b (n) A R (r, B) = np 2U (n) > r, W B = np [R > 2rU (n) P[W B [ = n d ɛ,n (2rU (n)) 1/η H (B) H (N ɛ) = r 1/η H (B) for r > {H (N)/n} ɛ η, which is precisely the decomposition of ν in (18). When examining the limiting measure of a set in the full subcone C which is not completely contained in C ɛ, a bias is induced by the choice of ɛ. To see this, extend the restricted hidden measure ia H{(, ɛ ɛ)} = H{(1 ɛ ɛ, 1)}, and consider a set A = [z, for z 1, z 2 >. Note that one can choose n and ɛ such that z C ɛ and z 1 + z 2 > {H (N)/n} ɛ η, and in this case we hae np [ Ê b (n) A = n = 1 1 b (n)z 1 w b (n)z 2 1 w { w z 1 1 w z 2 = (z 1 z 2 ) 1/2η 1 2 η 1 d ɛ,n r (1+1/η) drh ɛ (dw) } 1/η H (N ɛ )H ɛ (dw) = 1 ɛ ( ) 1/2η ɛ [z 1/η 1 + z 1/η 2 1 ɛ ɛ { w } 1/η 1 w H (dw) z 1 z 2 = ν ([z, ) B(ɛ, z), (21) 8

9 where the bias term B(, z) can be made arbitrarily small ia choice of. Figure 1 shows n = 25 simulated realizations of Z and Z for = 1 3, 1 2 and correlations of ρ =.8,.5,.2, as well as Z of De Haan and Zhou (211) and Y, the limiting first-order piece. Note that Z appears to capture the tail dependence structure of Z better than Z of De Haan and Zhou (211). The primary difference between Z simulations with =.1 and =.1 is the number of points near the axes of the cone C. As decreases we see more large points with angular components near and 1 in finite samples. This is due to the increase in scale parameter of R induced by smaller ; see Section 3.2. In Table 1, we proide a comparison of Z and Z by examining the empirical aerage number of points in specific sets oer 25 simulations of n = 25 points from both Z and Z for ρ =.5 and = 1 3, 1 2. Note that for =.1, the conergence to the limiting measure is slow for regions that are near the axes of the cone C. Howeer, for sets of the form A = [z,, choosing small results in near-unbiased estimation of ν (A) by Z. For sets that are near the axes, choosing slightly larger results in faster conergence to the limiting measure (in terms of z1, z2 ), particularly when considering marginal distributions. The trade-off is that the bias is greater for sets on C ; see Section 3.2. For comparison, we also show results from Z of De Haan and Zhou (211) and Y, the first-order piece of Z. We note that Z results in more points than expected in Z in most regions of the cone C. This is likely due to the slowly arying function (17), which is not accounted for by De Haan and Zhou. Of course, the first-order approximation Y fails to capture any of the distribution of Z on C z1 6 8 z1 8 6 z z z2 4 2 z1 Y Z* z z2 ^ (ε=.1) Z 8 ^ (ε=.1) Z Z z z1 Figure 1: Simulation of n = 25 points from (left to right) Z with Gaussian dependence structure (ρ =.5), Z = Y + E for =.1,.1, Z of De Haan and Zhou (211), and Y, the first-order limiting measure. Table 1: Summary statistics from 25 simulations of n = 25 points from Z, a biariate random ector with Fre chet marginals and Gaussian dependence with ρ =.5 and Z as constructed in Section 3.1 with = 1 3, 1 2. For comparison, we also show a summary of Z of De Haan and Zhou (211) and of Y, the first-order approximation to Z. Numbers reported are empirical means and simulation-based 95% interals. Number of points in the set [z, with z1 = z2 = z z Z Z ( =.1) Z ( =.1) Z Y (1, 6) 3.97 (1, 7) 2.89 (1, 6) (7, 16) 25.9 (, 3) 1.8 (, 3).84 (, 3) 3.21 (1, 6) 5.36 (, 2).36 (, 2).3 (, 2) 1.14 (, 3) Number of points with Z2 > z (2, 8) 9.39 (5, 14) 6.3 (2, 1) 8.48 (5, 14) 5. (2, 9) (, 6) 4.19 (2, 8) 2.86 (, 6) 3.97 (1, 7) 2.37 (, 5) (, 3) 1.89 (, 4) 1.42 (, 4) 1.93 (, 4) 1.16 (, 3) Choice of As Figure 1 and Table 1 indicate, one drawback of our construction Z is that it results in significantly more points near the axes than we see in Z. This is not surprising when one considers the limiting marginal measure of E : " # Z Z 1 1/η 1/η 1 w w Eˆ1 > z1 = H (N )H (dw) = H (dw) np b (n) z z 1 1 ( 1/2η 1/2η ) Z 1 1 1/η 1/η 1/η = z1 w H (dw) = z

10 For ery large z 1, this is negligible compared to the heaier-tailed Y 1 piece of Ẑ 1, which has limit measure z1 1. Howeer, for small ɛ the scaling factor in (22) is quite large, and plays a significant role in finite samples. This difficulty can be alleiated by choosing a slightly larger ɛ, which will reduce the magnitude of the scaling factor in (22). The drawback of choosing a larger alue for ɛ is that is increases the bias term in (21). That is, for sets in C for which smaller ɛ results in greater coerage by C ɛ, a larger ɛ increases the rate of conergence to the limiting measure, but also decreases the accuracy of the approximation to the limiting measure of such a set. Thus the choice of ɛ inoles a trade-off between the marginal behaior of Ẑ and the size of the restricted subcone C ɛ. While the infinite hidden angular measure of a Fréchet-marginal random ector with Gaussian dependence poses difficulty in simulation, our sum representation of Z in terms of independent Y and E proides seeral adantages oer preious approaches. We are able to capture not only the first-order limit on the whole cone C, but also the hidden regular arying piece on the subcone C ɛ. We can choose ɛ such that the restricted subcone C ɛ becomes arbitrarily close to C. Choosing ɛ inoles a trade off between bias in the limiting measure of sets not fully contained in C ɛ, and the leel at which the limiting measure is a good approximation for finite samples. We also point out here that our representation Y+E offers an adantage in finite samples oer a mixture of Y and E, as proposed by Maulik and Resnick (24). In applications in which marginal distributions are transformed to be unit Fréchet, no obseration has any component exactly equal to zero. This is also the case for our sum representation, but is not a feature of the mixture characterization. Finally, it is quite natural to think of tail obserations from a random ector Z as a sum of first- and second-order pieces. Viewing tail obserations as arising from a mixture distribution is not as intuitie. 4 Summary and Discussion This work presents a new representation of a multiariate regular arying random ector with hidden regular ariation, in terms of a sum of independent regular arying pieces. We hae shown our representation to be asymptotically justified ia the concept of multiariate tail equialence. An illustration of simulation from our model was proided using the biariate Gaussian as an example. The infinite hidden measure of this example introduced difficulty in simulation; howeer, we can still simulate the lighter-tailed piece on a restricted subcone. Our sum representation shares features with real data in applications and proides an intuitie model for the joint tail of a random ector. The sum representation proides a framework for maximum-likelihood estimation and likelihood-based model selection procedures. Finite samples of random ectors from random ectors exhibiting hidden regular ariation can be iewed as arising from the sum of components Y and E, whereas finite samples cannot be reconciled with the mixture representation of Maulik and Resnick (24). Our current work is to deelop appropriate statistical procedures. Specifically, as only realizations of Z are obsered, we treat Y and E as unobsered latent ariables and employ the EM algorithm (Dempster et al., 1977). One can write down a complete log-likelihood for a parameter ector θ goerning the tails of Y and E: l(θ z, y, e) = l Y (θ y) + l E (θ e) (22) based on limiting point process results for Y and E (Resnick, 27). Conditional on z and a fixed alue of the parameter ector θ (k), one can iew the conditional density of unobsered Y and E as p(y, e z, θ (k) ) = p Y(y θ (k) )p E (z y θ (k) ) p Z (z θ (k) ) p Y (y θ (k) )p E (z y θ (k) ). (23) One can speculate that in many cases, one can simulate realizations of Y and E from (23) without much difficulty. A Monte Carlo Expectation-Maximization algorithm could then be used to iteratiely compute and maximize Q(θ θ (k) ) = l(θ z, y, e)p(y, e z, θ (k) )dyde. (24) Estimation of this model ia the EM algorithm is a direction for future research. 1

11 Acknowledgements: The authors work has been partially funded by National Science Foundation grant DMS The authors also acknowledge support from the program on Uncertainty Quantification at the Statistical and Applied Mathematical Sciences Institute. GW also receied funding from the Weather and Climate Impacts Initiatie at the National Center for Atmospheric Research. References Ballani, F. and Schlather, M. (211). A construction principle for multiariate extreme alue distributions. Biometrika, 98(3): Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J., Waal, D. D., and Ferro, C. (24). Statistics of Extremes: Theory and Applications. Wiley, New York. Bingham, N., Goldie, C., and Teugels, J. (1989). Regular ariation, olume 27. Cambridge Uni Pr. Coles, S., Heffernan, J., and Tawn, J. (1999). Dependence measures for extreme alue analysis. Extremes, 2: Coles, S. and Tawn, J. (1991). Modeling multiariate extreme eents. Journal of the Royal Statistical Society, Series B, 53: Cooley, D., Dais, R., and Naeau, P. (21). The pairwise beta distribution: A flexible parametric multiariate model for extremes. Journal of Multiariate Analysis, 11(9): de Haan, L. (197). On regular ariation and its application to the weak conergence of sample extremes. Mathematisch Centrum. De Haan, L. and Zhou, C. (211). Extreme residual dependence for random ectors and processes. Adances in Applied Probability, 43(1): Dempster, A., Laird, N., and Rubin, D. (1977). Maximum likelihood from incomplete data ia the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), pages Draisma, G., Drees, H., Ferreira, A., and De Haan, L. (24). Biariate tail estimation: dependence in asymptotic independence. Bernoulli, 1(2): Heffernan, J. and Resnick, S. (27). Limit laws for random ectors with an extreme component. The Annals of Applied Probability, 17(2): Heffernan, J. E. and Tawn, J. A. (24). A conditional approach for multiariate extreme alues. Journal of the Royal Statistical Society, Series B, 66: Jessen, A. and Mikosch, T. (26). Regularly arying functions. Uniersity of Copenhagen, laboratory of Actuarial Mathematics. Ledford, A. and Tawn, J. (1997). Modelling depedence within joint tail regions. Journal of the Royal Statistical Society, Series B, B: Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multiariate extreme alues. Biometrika, 83: Maulik, K. and Resnick, S. (24). Characterizations and examples of hidden regular ariation. Extremes, 7(1): Mitra, A. and Resnick, S. (21). Hidden regular ariation: Detection and estimation. Arxi preprint arxi: Peng, L. (1999). Estimation of the coefficient of tail dependence in biariate extremes. Statistics and Probability Letters, 43: Ramos, A. and Ledford, A. (29). A new class of models for biariate joint tails. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(1):

12 Resnick, S. (22). Hidden regular ariation, second order regular ariation and asymptotic independence. Extremes, 5(4): Resnick, S. (27). Heay-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York. Sibuya, M. (196). Biariate extremal distribution. Ann. Inst. Statist. Math., 11: