Hidden Regular Variation under Full and Strong
Asymptotic Dependence

arXiv:1602.01262v3 [] 1 Feb 2017

BIKRAMJIT DAS
Engineering Systems and Design,
Singapore University of Technology and Design,
Singapore 487372
Email: bikram@sutd.edu.sg

and
SIDNEY I. RESNICK
School of ORIE,
Cornell University,
Ithaca, NY 14853 USA
Email: sir1@cornell.edu
Data exhibiting heavy-tails in one or more dimensions is often studied using the framework of regular
variation. In a multivariate setting this requires identifying specific forms of dependence in the data; this
means identifying that the data tends to concentrate along particular directions and does not cover the
full space. This is observed in various data sets from finance, insurance, network traffic, social networks,
etc. In this paper we discuss the notions of full and strong asymptotic dependence for bivariate data along
with the idea of hidden regular variation in these cases. In a risk analysis setting, this leads to improved
risk estimation accuracy when regular methods provide a zero estimate of risk. Analyses of both real and
simulated data sets illustrate concepts of generation and detection of such models.
28A33, 60G70, 62G05, 62G32.
Keywords: regular variation, multivariate heavy tails, hidden regular variation, tail estimation, strong dependence.

1. Introduction
Data that may be modeled by distributions having heavy tails appear in many contexts, for example,
hydrology ([1]), finance ([31]), insurance ([15]), Internet traffic ([5]), social networks and random
graphs ([4; 14; 28; 30]) and risk management ([7; 21]). Empirical evidence often indicates heavytailed marginal distributions and the dependence structure between the various components must
be discerned. We focus here on the case where components are strongly dependent.
The purpose of this paper is twofold. First, the paper encourages a definition of strong asymptotic
or extremal dependence that means the limit measure of regular variation concentrates on a cone
smaller than the full state space. Thus, directions where multivariate data from such a model are
found fall in a restricted set. Secondly, the paper shows that strong asymptotic dependence is a
tractable case for the applicability of hidden regular variation.
Hidden regular variation (HRV) [8; 11; 23–25; 27] is often considered for multivariate data exhibiting heavy tails when asymptotic independence is present. Asymptotic independence in a bivariate
1
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Das and Resnick

data set of positive values implies that both coordinates cannot be large simultaneously and therefore the multivariate regular variation (MRV) limit measure concentrates on the co-ordinate axes.
To improve risk estimation, one then seeks HRV on the non-negative orthant after removing the
two axes.
However, hidden regular variation is applicable whenever the limit measure of regular variation
in standard scale concentrates on a cone which is smaller than the entire state space, and is not
restricted to the case of asymptotic independence; see [8; 23] for details. If the limit measure
of regular variation concentrates on a relatively small cone, a risk calculation of a region in the
complement of the support of the limit measure will yield an answer of zero and HRV has the
potential to produce positive estimates of such risks.
We distinguish two related cases:
1. Full asymptotic dependence: the MRV limit measure concentrates in standard scale on a single
diagonal ray.
2. Strong asymptotic dependence: the MRV limit measure in standard scale concentrates on a
relatively small cone about the diagonal. This case is illustrated by analyzing out- and indegree for Facebook wall posts and returns of Chevron vs Exxon. The variables in these
examples are highly dependent, but they are not fully asymptotically dependent. In our
experience, it is much easier to find examples of strong asymptotic dependence compared
with full asymptotic dependence.
We review and adapt general model generation and detection techniques based on the generalized
polar coordinate transform; see [25, p. 198], [8; 11; 23]). The model generation methods produce
tractable models and the methods are illustrated in Sections 4.1 and 4.2. The detection methods
show when regularly varying models are consistent with data. We apply the detection methods to
the data examples of strong asymptotic dependence.
The mathematical framework for the study of multivariate heavy tails is regular variation of
measures. The theory is flexible when given for closed subcones of metric spaces [23]; we specialize
to subcones of R2+ and R2 where statistical results are most readily exhibited. Statistical extensions
to higher dimensions are possible and require more sophisticated graphics. We list needed notation
in Section 2.1 for reference. The definitions of multivariate regular variation (MRV) and hidden
regular variation (HRV) are reviewed in Section 2.2 where general concepts are adapted for subcones
in two dimensions. Sections 2.3 and 3 give equivalent formulations in polar co-ordinates and discuss
the particular cases of strong and full dependence.
Section 3.6 gives techniques for detecting when data is consistent with a model exhibiting MRV
and HRV. These techniques rely on the fact that under broad conditions, if a vector X has a
multivariate regularly varying distribution on a cone C, then under a generalized polar coordinate
transformation (see (2.6)), the transformed vector satisfies a conditional extreme value (CEV)
model for which detection techniques exist from [10]. This methodology adds to the toolbox of one
dimensional techniques such as checking if one dimensional marginal distributions are heavy tailed
or checking whether one dimensional functions of the data vector such as the maximum and the
minimum component are heavy tailed. See [25, p. 326], [24]. In Section 4, we analyze real and simulated data and show that our estimation and detection techniques produce results consistent with
presence of both MRV with strong dependence and HRV. Section 6 presents concluding comments.
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3

2. Background on Regular Variation of Measures
We provide a brief review of the mathematical setup for multivariate regularly varying measures
with the notion of M-convergence. More detail is found in [8; 11; 19; 20; 23]. The notions of
hidden regular variation (HRV) and regular variation expressed by polar coordinate transforms
are discussed in Sections 2.2 and 2.3 with emphasis on cases of the strong asymptotic dependence.
Finally in Section 3.6 we discuss detection of HRV using the Hillish estimator.

2.1. Basic notation.
A summary of some notation and concepts are provided here. For this paper, we restrict to dimension d = 2 unless otherwise specified. We use bold letters to denote vectors, with capital letters
for random vectors and small letters for non-random vectors, e.g., y = (y1 , y2 ) ∈ R2 . We also define 0 = (0, 0) and ∞ = (∞, ∞). Vector operations are always understood component-wise, e.g.,
for vectors x and y, x 6 y means xi 6 yi for i = 1, 2. Some additional notation follows with
explanations that are amplified in subsequent sections. Detailed discussions are in the references.
RV β

Regularly varying functions with index β > 0; that is, functions f : R+ 7→ R+
satisfying limt→∞ f (tx)/f (t) = xβ , for x > 0. We can and do assume such functions
are continuous and strictly increasing. See [3; 12; 26].

E

R2+ \ {0} or R2 \ {0}.

[diag]

{(x, x) : x > 0}.

[wedge]

{x ∈ R2+ : al x1 6 x2 6 au x1 } for some 0 < al < au < ∞.

να (·)

The Pareto measure on (0, ∞) given by να (x, ∞) = x−α , x > 0.

MRV

Multivariate regular variation; for this paper, it means regular variation on E.

HRV

Hidden regular variation; for this paper, it means a second regular variation after
removal of a cone as well as 0.

M(C \ C0 )

The set of all non-zero measures on C \ C0 which are finite on subsets bounded
away from the forbidden zone C0 , a closed cone removed from the state space.

C(C \ C0 )

Continuous, bounded, positive functions on C \ C0 whose supports are bounded
away from the forbidden zone C0 . Without loss of generality ([23]), we may assume
the functions are uniformly continuous.

µn → µ

Convergence in M(C \ C0 ) means µn (f ) → µ(f ) for all f ∈ C(C \ C0 ). See [8; 20; 23]
and Definition 2.1.
1/2
d(x, y)
Metric in R2 , usually the L2 distance d(x, y) = (x1 − y1 )2 + (x2 − y2 )2
.

x2
x1
diamond plot Mapping of thresholded data onto the L1 unit sphere x 7→ |x1 |+|x2 | , |x1 |+|x2 |

d(x, C)
ℵC

inf d(x, y) for x ∈ E and C ⊂ E.

y∈C

{x : d(x, C) = 1}. For instance: ℵ0 = {x ∈ E : d(x, {0}) = 1},
ℵ[diag] = {x ∈ E : d(x, [diag]) = 1} and ℵ[wedge] = {x ∈ E : d(x, [wedge]) = 1}.
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Das and Resnick
GPOLAR Generalized polar co-ordinate transformation relative to the deleted forbidden
zone C0 . GPOLAR(x) = (d(x, C0 ), x/d(x, C0 )) . See [8; 23].
X⊥
⊥Y

The random elements X and Y are independent.

2.2. Regularly varying distributions on cones.
We review material from [8; 20; 23] describing MRV and HRV specialized to two dimensions. The
convergence concept used for defining regular variation is M-convergence which is slightly different
from vague convergence traditionally used. Reasons for preferring M-convergence are discussed in
[8; 23].
2.2.1. Forbidden zones.
Consider R2+ or R2 as a metric space with Euclidean metric d(x, y). A subset C is a cone if it
is closed under positive scalar multiplication: if x ∈ C then c x ∈ C for c > 0. A framework for
discussing regularly varying measures is M-convergence ([8; 23]) on a closed cone C ⊂ R2+ or R2
with a closed cone C0 ⊂ C deleted. Call the deleted cone C0 the forbidden zone.
Here are some cases of interest for this paper; see Figure 1.
1. Suppose C = R2+ and C0 = {0}. Then E := C \ C0 = R2+ \ {0} is the space for defining
M-convergence appropriate for regular variation of distributions of positive random vectors.
The forbidden zone is the origin {0}.
2. Suppose C = R2 and C0 = {0}. Then E := C \ C0 = R2 \ {0} is the space appropriate
for regular variation of distributions of pairs of real valued random variables such as those
representing financial returns. The forbidden zone is still the origin {0}.
3. Suppose C = R2+ and C0 = {(x, x) : x > 0} =: [diag]. Then C \ C0 = R2+ \ [diag], the first
quadrant without its diagonal, is the right space for defining M-convergence appropriate for
HRV when asymptotic full dependence is present. The forbidden zone is the diagonal; see
Figure 1.
4. A related example is C = R2 and C0 = {(x, x) : x ∈ R} and we seek regular variation on
C \ C0 = R2 \ {(x, x) : x ∈ R}.
5. Suppose C = R2+ and for 0 < θl < θu < 1, and
0 < al = θu−1 − 1 < au = θl−1 − 1 < ∞,
x1
6 θu < 1}
x1 + x2
={x ∈ R2+ : al x1 6 x2 6 au x1 } =: [wedge],

C0 ={x ∈ R2+ : 0 < θl 6

(2.1)
(2.2)
(2.3)

where [wedge] is a pizza slice removed from the first quadrant. We then seek hidden regular
variation on R2+ \ [wedge]; see Figure 1. When al = au = 1 (or equivalently θl = θu = 1/2),
then [wedge] reduces to [diag]. Note we may parameterize [wedge] in two ways, one using the
slopes al , au and one using angles θl , θu . In (2.2) we use the traditional L1 polar coordinate
transform POLAR from R2 \ {0} 7→ (0, ∞) × [−1, 1] given by


x1
POLAR : x 7→ |x1 | + |x2 |,
= (r, θ).
|x1 | + |x2 |
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Strong Asymptotic Dependence

5

to express [wedge] in polar coordinates as R+ × [θl , θu ]. In (2.3) we use the slopes al , au of the
boundary lines of [wedge]. In practice we try to infer [wedge] by making a diamond plot of
the data using L1 norm thresholding.
In this paper we give particular attention to [wedge] because
• when the limit measure of regular variation concentrates on [wedge] ( R2+ we have a tractable
notion of strong asymptotic dependence; and
• data examples of strong asymptotic dependence seem to be far more common than for the
case of full asymptotic dependence.
Of course, other types of forbidden zones are possible and to date most attention has been directed
to removing axes when asymptotic independence is present.
2.2.2. Regular variation of measures.
Let M(C \ C0 ) be the set of Borel measures on C \ C0 which are finite on sets bounded away from
the forbidden zone C0 ([8; 20; 23]). We think of sets bounded away from the forbidden zone C0 as
tail regions. M-convergence is the basis for the definition of multivariate regular variation:
R
R
Definition 2.1. For µn , µ ∈ M(C \ C0 ) we say µn → µ in M(C \ C0 ) if f dµn → f dµ for all
f ∈ C(C \ C0 ).
Definition 2.2. A random vector Z > 0 is regularly varying on C \ C0 with index α > 0 if there
exists b(t) ∈ RV 1/α , called the scaling function, and a measure ν(·) ∈ M(C \ C0 ), called the limit
or tail measure, such that as t → ∞,
t P[Z/b(t) ∈ · ] → ν(·),

in M(C \ C0 ).

(2.4)

We write Z ∈ MRV(α, b(t), ν, C \ C0 ) to emphasize that regular variation depends on an index
α, scaling function b ∈ RV 1/α , limit measure ν, and state space C \ C0 . Since b(t) ∈ RV 1/α , the
limit measure ν(·) has a scaling property,
ν(c ·) = c−α ν(·),

c > 0.

(2.5)

Suppose C = R2+ , C0 = {0}. We distinguish between different forms of dependence and identify
them as follows:
1. If ν(·) satisfies ν((0, ∞)2 ) = 0 so that ν concentrates on the axes, then Z possesses asymptotic
independence; see [12; 25; 26].
2. If ν(·) concentrates on [diag] then Z has full asymptotic dependence.
3. If ν(·) concentrates on a narrow wedge as in (2.3), then Z has strong asymptotic dependence.
An analogous classification can be made for the case C = R2 .
Diamond plot: When doing empirical analyses for cases 2 or 3, it is convenient and informative
to map points


x1
x2
x 7→
,
= θ = (θ1 , θ2 )
|x1 | + |x2 |x1 | + |x2 |
onto the L1 unit sphere or diamond, perhaps after thresholding data according to the L1 norm.
We call the resulting plot the diamond plot. Observing how points cluster on the L1 unit sphere
provides a visualization of dependence.
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Das and Resnick

2.3. Regular variation and the polar coordinate transformation.
When the forbidden zone is the origin, it is useful to rephrase regular variation of measures using
the polar coordinate transformation. Theoretically, we may choose any norm k · k and the polar co
ordinate transform maps x into the unit sphere determined by the chosen norm: x 7→ kxk, x/kxk .
The limit measure expressed in polar coordinates is a product measure and this provides a way to
construct regularly varying measures and is useful for inference. (See [25, p. 168 ff, 173 ff].) When
the forbidden zone is a more general cone than just the origin, the polar coordinate transform no
longer brings benefits and the limit angular measure expressed in these coordinates may be infinite. To get a limit measure expressed as a product where the analogue of the angular measure
is a probability measure, one may transform (2.4) and (2.5) using generalized polar coordinates
([8; 23]). We can define the generalized polar coordinate transform for general cones of the form
C \ C0 and an associated metric d(·, ·) satisfying d(cx, cy) = cd(x, y) for scalars c > 0. The metric
d(·, ·) that we use in practice is the usual L2 Euclidean metric but note that the L1 norm is used
for visualizations using the diamond plot.
2.3.1. Generalized polar coordinates.
Define GPOLAR : C \ C0 7→ (0, ∞) × ℵC0 by
GPOLAR(x) =


d(x, C0 ),

x
d(x, C0 )


.

(2.6)

Consequently, the inverse GPOLAR← : (0, ∞) × ℵC0 7→ C \ C0 of the GPOLAR function is
GPOLAR← (r, θ) = rθ.

(2.7)

The transformation GPOLAR depends on the forbidden zone C0 and the choice of metric. In
practice, the metric d(·, ·) is taken to be the usual L2 Euclidean distance. This is practical and
customary but not obligatory.
2.3.2. Generalized unit sphere.
When transforming from Cartesian to polar coordinates, a central role is played by the unit sphere
ℵ0 := {x 6= 0 : kxk = 1}. The comparable set when using generalized polar coordinates with
respect to the forbidden zone C0 is ℵC0 = {x ∈ C \ C0 : d(x, C0 ) = 1}, the locus of points at
distance 1 from the deleted forbidden zone C0 . We then have an equivalent form of (2.4) and (2.5),
namely,


tP GPOLAR

Z
b(t)



h d(Z, C )

i
Z
0
,
∈ · → (να × S0 )(·) = (ν ◦ GPOLAR← )(·) ,
b(t)
d(Z, C0 )
(2.8)
in M ((0, ∞) × ℵC0 ) where να (x, ∞) = x−α , x > 0, α > 0 and S0 (·) is a probability measure on ℵC0
([8; 23]), provided b(t) is appropriately chosen. Note that ℵC0 depends on the choice of d(·, ·), and
the limit in (2.8) is a product measure.
h

∈ ·

i

= tP

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Strong Asymptotic Dependence

7

Figure 1. Left: R2+ \ {0} and distance is L2 . Center: R2+ \ [diag] and distance is L2 . Right: R2+ \ [wedge] and distance
is L2 . The dotted lines represent ℵC0 which is the locus of points at distance one from C0 .

2.3.3. Examples of unit spheres.
Figure 1 shows different shapes of ℵC0 for L2 distance and different choices of C0 where C = R2+ .
(i) For R2+ \ {0}, where the forbidden zone is {0}, we have ℵ0 = {x ∈ R2+ : d(x, {0}) = 1}.
(ii) If we delete the forbidden zone C0 = [diag] = {(x, x) : x ∈ R+ } from R2+ , the appropriate unit
sphere with respect to L2 distance is
ℵ[diag] := ℵ>[diag] ∪ ℵ<[diag] = {(u, u +

√

2) : u > 0}

[
√
√
{(u, u − 2) : u > 2},

(2.9)

the lines of slope 1, above and below the diagonal, which are at distance 1 from the diagonal.
For x ∈
/ [diag],
√

d x, [diag] = |x1 − x2 |/ 2.
(2.10)
(iii) When the forbidden zone is [wedge] ⊂ R2+ , we have
ℵ[wedge] =ℵ>[wedge] ∪ ℵ<[wedge]
q
q
[
p
={(u, au u + 1 + a2u ) : u > 0} {(u, al u − 1 + a2l ) : u > a−1
1 + a2l },
l

(2.11)

which are the lines parallel to the two rays defining [wedge] at a distance of 1 from [wedge].
When al = au = 1, ℵ[wedge] reduces to ℵ[diag] . For the distance to the forbidden zone from a
point x ∈
/ [wedge], we have
 |x2 − au x1 |
,
d x, [wedge] = p
1 + a2u
 |x2 − al x1 |
d x, [wedge] = q
,
1 + a2l

if x2 > au x1 ,

(2.12)

if x2 < al x1 ,

(2.13)

which reduces to (2.10) if al = au = 1.
Obvious changes apply when the state space is R2 .
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Das and Resnick

3. MRV and HRV under strong asymptotic dependence.
3.1. Definitions.
Consider simultaneous existence of regular variation on both the big cone R2+ \ {0} and a smaller
cone R2+ \C0 , where C0 is either [diag] or [wedge]. We provide equivalent polar-coordinate conditions
for this simultaneous existence.
Definition 3.1. The vector Z is regularly varying on R2+ \ {0} and has hidden regular variation
on R2+ \ C0 if there exist 0 < α 6 α0 , scaling functions b(t) ∈ RV 1/α and b0 (t) ∈ RV 1/α0 with
b(t)/b0 (t) → ∞ and limit measures ν, ν0 such that
Z ∈ MRV(α, b(t), ν, R2+ \ {0}) ∩ MRV(α0 , b0 (t), ν0 , R2+ \ C0 ).

(3.1)

Unpacking the notation we obtain the two regular variation limits
tP[Z/b(t) ∈ · ] → ν(·)
tP[Z/b0 (t) ∈ · ] → ν0 (·)

in M(R2+ \ {0}),
in

M(R2+

\ C0 ).

(3.2)
(3.3)

Using polar coordinates, (3.2) can be written as
tP [(kZk/b(t), Z/kZk) ∈ · ] → να × S(·)

in M((0, ∞) × ℵ0 ),

(3.4)

where S is a probability measure on ℵ0 . Similarly when removing C0 from the state space, generalized polar coordinates allow re-writing (3.3) as
tP

h d(Z, C )

i
Z
0
,
∈ · → (να0 × S0 )(·)
b0 (t)
d(Z, C0 )

(3.5)

in M ((0, ∞) × ℵC0 ) where να0 (x, ∞) = x−α0 , x > 0, α0 > 0 and S0 (·) is a probability measure on
ℵC0 .

3.2. Regular variation when deleting [wedge].
Focus on the special case where the forbidden zone is [wedge]. Since [diag] is a particular case of
[wedge], we do not treat [diag] separately. Recall the notation in (2.11) and the two parameterizations
of [wedge] given in (2.1), (2.2) and (2.3). The distance of points to [wedge] is given in (2.12) and
(2.13). When C0 = [wedge], (3.5) becomes two statements. With x > 0 and Λ ⊂ ℵ[wedge] we have
#
p
1 + a2u Z
Z2 − au Z1
p
> x,
∈ Λ →x−α0 S0 (Λ)
tP
Z2 − au Z1
b0 (t) 1 + a2u
q


2Z
1
+
a
a
Z
−
Z
l
2
l 1
q
tP 
> x,
∈ Λ →x−α0 S0 (Λ)
a
Z
−
Z
2
2
l 1
b0 (t) 1 + al
"


in M (0, ∞) × ℵ>[wedge] ,

(3.6)


in M (0, ∞) × ℵ<[wedge] .

(3.7)

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Strong Asymptotic Dependence

9

Modifying variables leads to the two simpler statements. For x > 0,
(
)
p




2
p
α
1
+
a
Z2 − au Z1
Z2
u
2 20 −α0
tP
6 y →(1 + au ) x
, y > au ,
> x,
S0 v, au v + 1 + a2u : v >
b0 (t)
Z1
y − au
(3.8)
q


q
n
1 + a2l o

α
al Z1 − Z2
Z1
1
2 20 −α0
2
tP
6 y →(1 + al ) x
S0 v + 1 + al /al , v : v >
> x,
, y> .
b0 (t)
Z2
al y − 1
al
(3.9)
Remark 3.1. Thus a necessary condition for non-trivial regular variation on R2+ \ [wedge] is that
both (Z2 − au Z1 )+ and (al Z1 − Z2 )+ be regularly varying with index α0 > α. This fact suggests
the exploratory diagnostic of testing whether these 1 dimensional variables have power laws with
the same index. If so, one can continue to explore with the Hillish statistic and associated plot; see
Section 3.6.
However, there is nothing to prevent the possibility that regular variation exists on the region
above [wedge] but that tails are of lower order below [wedge]. This would happen for instance if
S0 (ℵ<[wedge] ) = 0 but S0 (ℵ>[wedge] ) > 0. If this happened, one could search for another higher index
or thinner tailed regular variation on < [wedge].
Remark 3.2. Analogous statements to the limits (3.8), (3.9) hold true when C = R2 and [diag]
and [wedge] are their appropriate equivalents in R2 to get.
3.2.1. Restrictions on the choice of [wedge].
In this paper, we assume that 1 < al 6 1 6 au < ∞ where [wedge] = {x ∈ R2+ : al x1 6 x2 6 al x1 }.
This choice is partly governed by the fact that it is easier for us to deal with data portraying tail
equivalence with limt→∞ P(Z1 > t)/P(Z2 > t) = 1. Now, when al = au , we get [wedge] = [diag],
which means under a model of full asymptotic dependence the only limit measure that we allow is
restricted to [diag]. If we assume Z ∈ MRV(α, b(t), ν, R2+ \ {0}) and ν is supported on [diag] then
using (3.2), this clearly implies, that
lim

t→∞

P(Z1 > b(t))
P(Z1 > t)
= lim
P(Z2 > t) t→∞ P(Z2 > b(t))
t P (Z/b(t) ∈ (1, ∞) × [0, ∞))
= lim
t→∞ t P (Z/b(t) ∈ [0, ∞) × (1, ∞))
ν ((1, ∞) × [0, ∞))
=
ν ([0, ∞) × (1, ∞))
ν ((x1 , x2 ) : x1 > 1, (x1 , x2 ) ∈ [diag])
=
= 1,
ν ((x1 , x2 ) : x2 > 1, (x1 , x2 ) ∈ [diag])

where the last line is a consequence of ν being concentrated on [diag]. Thus, not only are Z1 , Z2
tail equivalent, but in fact limt→∞ P(Z1 > t)/P(Z2 > t) = 1.
The following lemma shows that that we cannot choose a [wedge] which does not contain [diag]
if we want to guarantee limt→∞ P(Z1 > t)/P(Z2 > t) = 1.
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Das and Resnick

Lemma 3.1. If Z ∈ MRV(α, b(t), ν, R2+ \ {0}) where ν is supported on [wedge] = {x ∈ R2+ :
al x1 6 x2 6 al x1 } and limt→∞ P(Z1 > t)/P(Z2 > t) = 1 then 0 < al 6 1 6 au < ∞.
Proof. To get a contradiction, suppose we have 1 < al 6 au < ∞. (A similar contradiction is
obtained if we assume 0 < al 6 au < 1.) Assume,
P(Z1 > t)
t P (Z/b(t) ∈ (1, ∞) × [0, ∞))
ν ((1, ∞) × [0, ∞))
= lim
=
.
t→∞ P(Z2 > t)
t→∞ t P (Z/b(t) ∈ [0, ∞) × (1, ∞))
ν ([0, ∞) × (1, ∞))

1 = lim

Hence ν ((1, ∞) × [0, ∞)) = ν ([0, ∞) × (1, ∞)). We know that ν is supported on [wedge]. So,
ν ((1, ∞) × [0, ∞)) = ν((x1 , x2 ) ∈ R2+ : x1 > 1, (x1 , x2 ) ∈ [wedge])


x2
2
6 au
= ν (x1 , x2 ) ∈ R+ : x1 > 1, al 6
x1


1
x2
2
< ν (x1 , x2 ) ∈ R+ : x1 > , al 6
6 au
al
x1


x2
< ν (x1 , x2 ) ∈ R2+ : x2 > 1, al 6
6 au
x1
= ν ([0, ∞) × (1, ∞)) ,

(since al > 1)
(since x2 > al x1 )

which is a contradiction.

3.3. Regular variation when deleting [wedge] expressed in traditional polar
coordinates.
Regular variation on R2+ \ [wedge] expressed using generalized polar coordinates
 in (3.8), (3.9) can
also be written in terms of the traditional polar coordinates z 7→ r, (θ, 1 − θ) where z1 = rθ and
z2 = r(1 − θ) and r = z1 + z2 . Using capital letters for random variables, the left most probability
in (3.8) becomes for x > 0, y > au ,


hZ − a Z
i
Z2
R(1 − Θ(1 + au ))
(1 − Θ)
2
u 1
tP
> x,
6 y = tP
> x,
6y
b0 (t)
Z1
b0 (t)
Θ
and using (2.1) this is
#
"
#
R(1 − θl−1 Θ)
R(1 − θl−1 Θ)
1
−1
=tP
> x, Θ 6 y + 1 = tP
> x, Θ >
b0 (t)
b0 (t)
1+y
"

Set s = 1/(1 + y) where y > au and thus s < θl ,
"
#
(
)
p


2
p
α0
R(1 − θl−1 Θ)
1
+
a
u
tP
.
> x, Θ > s → (1 + a2u ) 2 x−α0 S0 v, au v + 1 + a2u : v > −1
b0 (t)
s − θl−1
An analogous expression holds for (3.9). So if regular variation with index α holds on R2+ \ {0},
R is a random variable with regularly varying distribution tail with index α and multiplying R by
1 − θl−1 Θ produces a variable with a lighter tail having index α0 .
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

11

3.4. The forbidden zone of HRV and the limit measure on E.
Suppose there are two regular variation properties that hold for a vector Z > 0 so that (3.1)
holds with b(t)/b0 (t) → ∞. If C0 = [wedge] with 0 < al 6 1 6 au < ∞; then ν, the limit
measure on R2+ \ {0}, must concentrate on the forbidden zone [wedge] used to define the second
regular variation. (Cf. [25, p. 324-5].) This means that when detecting MRV, if the limit measure
is typical of strong asymptotic dependence and concentrates on [wedge], we are encouraged to look
for additional regular variation regimes on R2+ \ [wedge]. Hence we get the following result.
Proposition 3.1.

Suppose

Z ∈ MRV(α, b(t), ν, R2+ \ {0}) ∩ MRV(α0 , b0 (t), ν0 , R2+ \ [wedge])

(3.10)

with b(t)/b0 (t) → ∞ and 0<al 6 1 6 au < ∞. Then ν, the limit measure on R2+ \ {0}, concentrates
on [wedge].
Proof. To see this, consider the region above the ray y = au x. Then for δ > 0, as t → ∞, referring
to (3.8),
hZ − a Z
i
hZ − a Z
b(t) i
2
u 1
2
u 1
tP
> δ =tP
>
δ → 0,
b(t)
b0 (t)
b0 (t)
since b(t)/b0 (t) → ∞ and S0 (·) is a probability measure. Similarly, for the region below the ray
y = al x,
ha Z − Z
i
ha Z − Z
b(t) i
2
2
l 1
l 1
tP
> δ =tP
>
δ → 0,
b(t)
b0 (t)
b0 (t)
from (3.9). Thus ν places no mass outside [wedge].
Clearly, a result analogous to Proposition 3.1 holds where C = R2 and C0 is the appropriate
equivalent of [diag] and [wedge] on R2 .

3.5. How HRV on R2 \ [wedge] can improve risk estimates.
Suppose I1 and I2 are financial instruments that have positive risks Z1 and Z2 per unit of investment
where Z = (Z1 , Z2 ) satisfies (3.10). Suppose we buy one unit of I2 and sell 2al units of I1 . The risk
of this portfolio is Z2 − 2al Z1 and we have two asymptotic regimes that can be used to estimate
the probability the risk is large. If we use MRV with scale function b(t) then for large x > 0,
h Z
i
Z2 
1
P[Z2 − 2au Z1 > x] =P
,
∈ {(v, w) ∈ R2+ : w − 2au v > x/b(t)}
b(t) b(t)
1
≈ ν{(v, w) ∈ R2+ : w − 2au v > x/b(t)} = 0
t
since the required region is outside the support [wedge] of the measure ν. Is the risk really 0 or did
we use the wrong asymptotic approximation? If we use HRV with scale function b0 (t), then we get
a non-zero limit:
h Z
i
Z2 
1
P[Z2 − 2au Z1 > x] =P
,
∈ {(v, w) ∈ R2+ : w − 2au v > x/b0 (t)}
b0 (t) b0 (t)
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

12

Das and Resnick
1
≈ ν0 {(v, w) ∈ R2+ : w − 2au v > x/b0 (t)}.
t

Now switch to generalized polar coordinates with (v, w) = r(µ1 , µ2 ) and µ2 = au µ1 and the risk
calculation is with respect to the product measure να0 × S0 d(µ1 , µ2 ) and
ZZ

1
α0 r−α0 −1 S0 d(µ1 , µ2 )
t
{(r,(µ1 ,µ2 )):rµ2 −2au rµ1 >x/b0 (t)}
Z

1  x −α0
=
(µ2 − 2au µ1 )α0 S0 d(µ1 , µ2 ) .
t b0 (t)
{(µ1 ,µ2 ):µ2 −2au µ1 >0}

P[Z2 − 2au Z1 > x] ≈

Of course, in practice S0 , α0 , b0 must be replaced by estimators and t is replaced by n/k where n is
the sample size of observations and k is the number of observations used in estimation. An example
where we carry out these calculations is given for simulated data in Section 4.2.1.

3.6. Exploring for HRV with the Hillish estimator.
The Hillish estimator was designed for detection of the CEV model [8–10; 16–18; 23] and extended
to detecting hidden regular variation in [11]. The generalized polar coordinate transform converts
Cartesian coordinates in the definition of regular variation into coordinates satifying the CEV
model. In this paper we show that the Hillish technique can detect HRV when the cone removed
from R2+ or R2 is [wedge]. The Hillish procedure is described below. First we define a conditional
extreme value model.
3.6.1. The CEV model.
Suppose the random variables (ξ, η) form a random element of R+ × R and there exists a regularly
varying function b(t) → ∞ and a non-null measure µ ∈ M((0, ∞) × R) and
tP

h ξ

i
, η ∈ · → µ(·),
b(t)

in M((0, ∞) × R).

(3.11)

Note that (2.8) is of this form where only the first component ξ = d(Z, C0 ) is scaled. See also (3.8)
and (3.9). Additionally assume that
(a) µ((r, ∞] × · ) is a non-degenerate measure for any fixed r > 0, and,
(b) H(·) := µ((1, ∞) × · ) is a probability distribution.
Then (ξ, η) satisfies a conditional extreme value model and we write (ξ, η) ∈ CEV(b, µ). Note
(3.8) and (3.9) are of the form given in (3.11). Hence the HRV statements are equivalent to the
appropriate transforms of the variables following a CEV model.
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

13

3.6.2. The Hillish procedure.
Now suppose (ξi , ηi ); 1 6 i 6 n are iid replicates of (ξ, η). Define
ξ(1) > . . . > ξ(n)

The decreasing order statistics of ξ1 , . . . , ξn .

ηi∗ ,

The η-variable corresponding to ξ(i) , also called

16i6n

the concomitant of ξ(i) .
Nik =

k
P
l=i

1{ηl∗ 6ηi∗ }

Rank of ηi∗ among η1∗ , . . . , ηk∗ . We write Ni = Nik .

∗ 6 η∗ 6 . . . 6 η∗
∗
∗
η1:k
2:k
k:k The increasing order statistics of η1 , . . . , ηk .

By analogy with the Hill estimator and the Hill plot, the Hillish statistic is defined for 1 6 k 6 n
as
Hillishk,n = Hillishk,n (ξ, η) :=

k
1X
k
k
log log k
k
j
Nj
j=1

(3.12)

According to [10, Propositions 2.2 and 2.3], if (ξ, η) ∈ CEV(b, µ) then there exists a limit Iµ and
P

Hillishk,n → Iµ and moreover, µ is a product measure iff both
P
Hillishk,n (ξ, η) →
1

and

P
Hillishk,n (ξ, −η) →
1,

(3.13)

as k → ∞, n → ∞, n/k → ∞. Note the limits in (3.8), (3.9) are all product measures. Hence the
diagnostic for detecting regular variation with a specified forbidden zone for the random vector Z
is to plot the Hillish statistic of GPOLAR(Z).
We emphasize that if (3.13) holds, we have empirical behavior consistent with the presence of
regular variation but this does not prove existence of regular variation. When the Hillish technique
fails because the plots do not hug the line at height 1, we can reject a hypothesis of regular variation.

4. Data Analysis with Simulated Data
Before moving to real data, we test our analysis techniques on two simulated data sets to see how
well they perform in Section 4.1 and 4.2. Further, we discuss MRV and HRV properties and their
detection. In Section 5, we analyze two real bivariate data sets both of which exhibit heavy-tailed
margins and strong asymptotic dependence.

4.1. Example 1: Full asymptotic dependence.
Suppose Z1 ∼ Pareto(1.5) and Z2 ∼ Pareto(2.5) and independent of each other. Let B1 , B2 be
iid Bernoulli (0.5) random variables also independent of Z1 and Z2 . Now define the vector X =
(X1 , X2 ) as
X1 = B1 Z1 + (1 − B1 )Z2 ,
X2 = B1 Z1 + B2 (1 − B1 )(1.5Z2 ) + (1 − B2 )(1 − B1 )(0.5Z2 )
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

14

Das and Resnick
Hill plot: data

Hill plot: data
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4

80

●

0

500

1000

1500

number of order statistics

2000

0

500

1000

1500

2000

number of order statistics

Figure 2. Example 4.1: (Left) Scatter plot of 10,000 data points. (Center and right) Hill plots for tail parameters of
the marginal distributions of X1 (center) and X2 (right).The horizonal lines are at height 1.5.

By construction


(Z1 , Z1 ),
X = (Z2 , 1.5Z2 ),


(Z2 , 0.5Z2 ),

with probability P [B1 = 1] = 12 ,
with probability P [B1 = 0, B2 = 1] = 41 ,
with probability P [B1 = 0, B2 = 0] = 41 ,

so that X = (X1 , X2 ) lies on y = x with probability 0.5. With probability 0.25 each it is either on
the line y = 0.5x or on y = 1.5x. In this model, X is MRV with parameter α = 1.5 and it has full
dependence on the diagonal [diag] := {(x, y) ∈ R2+ : y = x}. On the other hand, on R2+ \ [diag],




d((X1 , X2 ), [diag])
(X1 , X2 )
(2|X1 − X2 |
(X1 , X2 )
√
tP
> x, √
= a = tP
> x,
=a
|X1 − X2 |
t1/2.5
t1/2.5 /2 2
2d((X1 , X2 ), [diag])
√
1
1
→ x−2.5 × = ν2.5 (x, ∞)S0 ({ 2a}).
4
2
√
where a = (2, 1) or (2, 3) and S0 ( 2a) = 0.5. Hence we have HRV on R2+ \[diag] with tail parameter
α0 = 2.5.
We generate n = 10, 000 iid samples from this data set. The scatter plot in Figure 2 shows the
dependence structure of X along with Hill plots of X1 , X2 which supports the premise that α = 1.5.
To understand the dependence structure of the variables X, we make the diamond plot, the
transformation from R2 7→ ℵ0 ⊂ R2 onto the L1 unit sphere represented by the diamond {(θ1 , θ2 ) :
|θ1 | + |θu | = 1} We do the mapping at various thresholds determined by k, the number of order
statistics of the norms kxk = |x1 | + |x2 |. In Figure 3, the diamond plot and histogram of the
angular measure is shown for k = 100. Clearly the data is concentrated at x = y. The Hill plot
of the quantity |X1 − X2 | supports the fact that data was generated with hidden tail parameter
α0 = 2.5.
Finally, we look at the Hillish statistic for (ξ, η) = (|X1 − X2 |, X1 /|X1 − X2 |) after removing
[diag]. The Hillish plot in Figure 4 is convincingly stable and close to 1, supporting the presence of
HRV as expected from the generation procedure here.
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

15

Histogram of θ1

Diamond plot

y2

0.0
−0.5

0

0

−1.0

1

5

2

Hill estimate of alpha

0.5

15

●
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10

Density

●

3

4

20

1.0

5

Hill plot: data

0.0

0.2

0.4

0.6

0.8

1.0

−1.0

−0.5

θ1

0.0

0.5

1.0

0

500

y1

1000

1500

2000

number of order statistics

Figure 3. Example 4.1: Histogram of θ1 (left), diamond plot (center), and Hill plot for tail estimate of |X1 − X2 |
(right). The horizontal line is at height 2.5.

Hillish(R,− θ)
2.0
1.5
1.0

Hillish statistic

0.0

0.5

1.0
0.0

0.5

Hillish statistic

1.5

2.0

Hillish(R,θ)

0

1000

2000

3000

4000

5000

0

1000

number of order statistics

2000

3000

4000

5000

number of order statistics

Figure 4. Example 4.1: Hillish plots for (ξ, η) (left) and (ξ, −η) (right) where (ξ, η) = (|X1 − X2 |, X1 /|X1 − X2 |).

4.2. Example 2: Strong asymptotic dependence.
In this example, we simulate data from a model with strong asymptotic dependence. We also exhibit
estimation of rare probabilities and conduct a sensitivity analysis when the support of the regular
variation for the the first level, and hence that for HRV is not correctly identified.
Suppose R1 ∼ Pareto(1.5) and R2 ∼ Pareto(2.5) and independent of each other. Let Θ1 ∼
Unif[0.4, 0.6], Θ2 ∼ Unif([0, 1]\[0.4, 0.6)], B ∼ Bernoulli (0.5) random variables. Assume the random
variables are all independent. Now define the vector X = (X1 , X2 ) as
X1 = BR1 Θ1 + (1 − B)R2 Θ2 ,
X2 = BR1 (1 − Θ1 ) + (1 − B)R2 (1 − Θ2 ).
By construction, X is MRV on R2+ \ {0} with tail parameter α = 1.5. Corresponding to (θl , θu ) =
(0.4, 0.6), we have by (2.1) that (al , au ) = (0.67, 1.5) and therefore
[wedge] := {(x, y) ∈ R2+ : 0.67x 6 y 6 1.5x}.
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

16

Das and Resnick
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80

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0

500

x1

1000

1500

2000

0

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500

1000

1500

2000

number of order statistics

Figure 5. Example 4.2: (Left) Scatter plot of 30,000 data points. (Center and right) Hill plots for tail parameters of
the marginal distributions of X1 , X2 . The horizontal lines are at height 1.5.

This gives hidden regular variation with tail parameter α0 = 2.5 on R2+ \ [wedge].
Histogram of θ1

Diamond plot

●

y2

4

0.0
−0.5

3

Density

0

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Hill plot: data

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−0.5

0.0
y1

0.5

1.0

0

500

1000

1500

2000

number of order statistics

Figure 6. Example 4.2: Diamond plot, histogram and Hill plot for tail estimate of d(X, [wedge]). The horizontal line
in the Hill plot is at height 2.5.

We generate n = 30, 000 iid samples from this data set. The scatter plot in Figure 5 shows the
dependence structure of X along with Hill plots of X1 , X2 which supports the premise that α = 1.5.
To understand the dependence structure of X, we graph the diamond plot as used in the previous
example. We do the mapping at various thresholds determined by k, the number of order statistics
of the norms |x| + |y|. In Figure 6, the histogram of angles and the diamond plot are shown for
k = 100 and shows the angles are Uniform in [0.4, 0.6] for high values of |x| + |y|. The Hill plot of
the quantity d(X, [wedge]) supports the fact that data was generated with hidden tail parameter
α0 = 2.5. Finally, we look at a Hillish statistic for (ξ1 , η1 ) and (ξ2 , η2 ) respectively which are
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

17
Hillish(R,− θ)

2000

4000

6000

0

number of order statistics

2000

4000

6000

2.0
0.5
0.0
0

number of order statistics

1.0

Hillish statistic

1.5

2.0
0.0

0.5

1.0

Hillish statistic

1.5

2.0
1.5
1.0

Hillish statistic

0.0
0

Hillish(R,− θ)

Hillish(R,θ)

0.5

1.0
0.0

0.5

Hillish statistic

1.5

2.0

Hillish(R,θ)

2000

4000

6000

number of order statistics

0

2000

4000

6000

number of order statistics

Figure 7. Example 4.2: Hillish plots for (i)(ξ1 , η1 ), (ii)(ξ1 , −η1 ), (iii)(ξ2 , η2 ), (iv)(ξ2 , −η2 ) respectively where (ξ1 , η1 )
and (ξ2 , η2 ) are obtained by using (3.8) and (3.9) on X = (X1 , X2 ).

obtained by using (3.8) and (3.9) on X = (X1 , X2 ) after removing [wedge]. The Hillish plots in
Figure 7 are again convincingly stable and close to 1 and detect the hidden regular variation.
4.2.1. Probabilities of rare sets for this example.
Now to further illustrate our methods, we compute P(X2 −2X1 > x) and P(X2 −3X1 > x). Without
resorting to hidden regular variation we have X is MRV on R2+ \ {0} with tail parameter α = 1.5
and the limit measure concentrates on
[wedge] := {x ∈ R2+ : al x1 6 x2 6 au x1 } = {x ∈ R2+ : 0.67x1 6 x2 6 1.5x1 }.
Hence with the usual regular variation techniques we would estimate both
P(X2 − 2X1 > x) ≈ 0

and P(X2 − 3X1 > x) ≈ 0.

But for this example we can compute the exact answer without resorting to asymptotic approximations and we get,
1
1
p1 (x) :=P(X2 − 2X1 > x) = P(R1 (1 − Θ1 )−2R1 Θ1 > x) + P(R2 (1 − Θ2 )−2R2 Θ2 > x)
2
2
and because 3Θ1 > 1, this is
1
5
= P(R2 (1 − 3Θ2 ) > x) = x−2.5 .
2
84

(4.1)

Similarly we can compute
p2 (x) :=P(X2 − 3X1 > x) =

5 −2.5
x
.
112

(4.2)

imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

18

Das and Resnick
p1(4)

1.5
0.0

0.5

1.0

1.5
0.0

0.5

1.0

1.5
0.0

0.5

1.0

1.5
1.0
0.5
0.0

2.0

p1(4)
2.0

^ (4)
p
1

p1(1)
2.0

p1(4)

p1(1)
2.0

p1(1)

^ (1)
p
1

●

p2(1)

^ (1)
p
2
●

p2(4)

^ (4)
p
2

p2(1)

p2(1)

p2(4)

p2(4)
2.0

●

2.0

2.0

2.0

●

1.5
1.0
0.5

1.5
1.0
0.5

1.5
1.0
0.5

0.5

1.0

1.5

●

●

Figure 8. Example 4.2: Boxplots for

p̄1 (1) p̂1 (1) p̄1 (4) p̂1 (4)
,
,
,
p1 (1) p1 (1) p1 (1) p1 (4)

(top) and

0.0

0.0

0.0

0.0

●

p̄2 (1) p̂2 (1) p̄2 (4) p̂2 (4)
,
,
,
p2 (1) p2 (1) p2 (4) p2 (4)

(bottom) .

If we pretend we do not know the exact answers provided by (4.1), (4.2), can we give better
estimates than 0 using asymptotic methods on our simulated data set? Under hidden regular
variation after removing C0 = [wedge], we know that (3.5) holds for X and hence

i
n h d(X, C0 )
X
P
>
x,
∈
Λ
→ x−α0 × S0 (Λ)
k
b0 ( nk )
d(X, C0 )

(4.3)

as n → ∞, k → ∞, k/n → 0, where S0 (·) is a probability measure and Λ ⊂ ℵC0 . From Section 3.5,
we have for x > 0, as n → ∞, k → ∞, k/n → 0,
Z

n
n
n
n
−α0
p1 (b0 ( )x) = P[X2 − 2X1 > b0 ( )x] → x
(µ2 − 2µ1 )α0 S0 d(µ1 , µ2 ) ,
k
k
k
k
{(µ1 ,µ2 ):µ2 −2µ1 >0}
(4.4)
with a similar limiting expression for p2 (x). This suggests we need to estimate α0 , b0 (n/k), S0 (·)
and of course the wedge C0 .
For the wedge, we estimate âl , âu using the 5th and 95th percentile of the range of X1 /(X1 + X2 )
for 100 highest values of X1 + X2 for each simulation. In (4.3) replace x by 1 and Λ by ℵC0 and
then we estimate b0 (n/k) with the kth largest value of d(X i , C0 ) corresponding to X 0 s outside
C0 . Alternatively, if we fix b0 (n/k) then, we obtain the appropriate k largest value of d(X i , C0 )
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

19

0

500

1000

1500

2000

4
3
0

1

2

Hill estimate of alpha

3
0

1

2

Hill estimate of alpha

3
2
0

1

Hill estimate of alpha

Hill plot with al = 1, au = 1

4

Hill plot with al = 0.9, au = 1.11

4

Hill plot with al = 0.5, au = 2

0

500

number of order statistics

1000

1500

2000

0

500

number of order statistics

1000

1500

2000

number of order statistics

Figure 9. Example 4.2: Hill plots for estimating α0 (known to be 2.5 in the model) when the support set of MRV
is incorrectly specified as (al , au ) = (0.5, 2) (left), (al , au ) = (0.9, 1.11) (middle), (al , au ) = (1, 1) (right) respectively.
The horizontal line in the Hill plot is at height 2.5.

Hillish(R,− θ)

2000

6000

10000

0

number of order statistics

2000

6000

10000

2.0
0.5
0.0
0

number of order statistics

1.0

Hillish statistic

1.5

2.0
0.0

0.5

1.0

Hillish statistic

1.5

2.0
1.5
1.0

Hillish statistic

0.0
0

Hillish(R,− θ)

Hillish(R,θ)

0.5

1.0
0.0

0.5

Hillish statistic

1.5

2.0

Hillish(R,θ)

2000

6000

10000

0

number of order statistics

2000

6000

10000

number of order statistics

Figure 10. Example 4.2: Hillish plots for (i)(ξ1 , η1 ), (ii)(ξ1 , −η1 ), (iii)(ξ2 , η2 ), (iv)(ξ2 , −η2 ) respectively where (ξ1 , η1 )
and (ξ2 , η2 ) are obtained by using (3.8) and (3.9) on X = (X1 , X2 ) and the support of MRV is (incorrectly) identified
by (al , au ) = (0.9, 1.11).

corresponding to X 0 s outside C0 . to be used. For S0 (·) we modify the argument leading to [25, Eq.
9.47, p. 313].
b 0 , b̂0 (n/k), and choice of k, we enumerate the gpolar-transformed
Corresponding to estimates C
b
b 0 ) as {(ri , µ1i , µ2i ) : 1 6 i 6 k}.
points outside C0 , corresponding to the k largest values of d(X i , C
Then using these points we estimate S0 (·) with the empirical distribution as

Ŝ0 (·) =

k
1X
(µ1i ,µ2i ) (·).
k
i=1

This leads to the risk estimates,
k
k
1X
p̂1 (x) =x−α̂0 (b̂0 (n/k))αˆ0
(µ2i − 2µ1i )αˆ0 1{µ2i >2µ1i } ,
n
k

(4.5)

i=1

imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

20

Das and Resnick
^ (1)
p
1

p1(4)

^ ( 4)
p
1

p1(1)

p1(1)

p1(4)

p1(4)

2.0

2.0

2.0

●

2.0

p1(1)

●

●

1.5

1.5

1.5

1.5

●

1.0
0.5

1.0

1.0
0.5

0.5

1.0
0.5

●

0.0

0.0

0.0

p2(4)

^ ( 4)
p
2

p2(1)

p2(4)

p2(4)

●

2.0

^ (1)
p
2

p2(1)

2.0

p2(1)

2.0

2.0

0.0

●

●

Figure 11. Example 4.2: Boxplots for p̄p11 (1)
, p̂1 (1) , p̄1 (4) ,
(1) p1 (1) p1 (4)
correct) identification of support where al = 0.5, au = 2.

p̂1 (4)
p1 (4)

1.5
0.0

0.5

1.0

1.5
0.0

0.5

1.0

1.5
0.5
0.0

1.0

0.5
0.0

1.0

1.5

●

(top) and

p̄2 (1) p̂2 (1) p̄2 (4) p̂2 (4)
,
,
,
p2 (1) p2 (1) p2 (4) p2 (4)

k
k
1X
p̂2 (x) =x−α̂0 (b̂0 (n/k))αˆ0
(µ2i − 3µ1i )αˆ0 1{µ2i >2µ1i } .
n
k

(bottom) with (in-

(4.6)

i=1

Since α0 , al , au are known for this simulation example, we may compare p̂i (x) with p̄i (x) estimated
using the three known values. We carry out comparisons using x = 1 and x = 4. We conduct
simulations with n = 10, 000 and use a value of k corresponding to b0 (n/k) = 2. We compute
p̂1 (4) p̄1 (1) p̄1 (4)
p̂1 (1), p̂1 (4), p̄1 (1), p̄1 (4), for 100 iterations and create box plots of p̂p11 (1)
(1) , p1 (4) , p1 (1) , p1 (4) and we
do the same for p2 (1) and p2 (4). From Figure 8, clearly the estimates perform pretty well since the
ratios of the estimates to the real values are very close to 1. Clearly, when α0 is estimated the error
bounds become larger, but still perform reasonably. Note that the quantities we compute have low
probabilities:
p1 (1) = 0.0595,

p2 (1) = 0.044,

p1 (4) = 0.002,

p2 (4) = 0.0014.

To summarize: This estimation procedure can be used to calculate risk probabilities in the
presence of hidden regular variation when the primary regular variation gives a zero risk estimate.
4.2.2. Sensitivity analysis in this example
Clearly, the probability estimation procedure we discussed hinges on our ability to appropriately
estimate the support set of regular variation at the first level, given by
[wedge] := {x ∈ R2+ : al x1 6 x2 6 au x1 }.
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

21

p1(1)

p1(1)

p1(4)

p1(4)

●

2.0

^ ( 4)
p
1

2.0

p1(4)

2.0

^ (1)
p
1

2.0

p1(1)

●

●

1.5
1.0

1.5
1.0

1.5
1.0

1.0

1.5

●
●
●

0.5
0.0

0.5
0.0

0.5
0.0

p2(4)

^ ( 4)
p
2

p2(1)

p2(4)

p2(4)
2.0

^ (1)
p
2

p2(1)

2.0

p2(1)

2.0

2.0

0.0

0.5

●

●
●
●

Figure 12. Example 4.2: Boxplots for p̄p11 (1)
, p̂1 (1) , p̄1 (4) , p̂1 (4) (top) and
(1) p1 (1) p1 (1) p1 (4)
correct) identification of support where al = 0.9, au = 1.1.

1.5
1.0
0.5
0.0

1.5
0.5
0.0

1.0

1.5
0.5
0.0

1.0

0.5
0.0

1.0

1.5

●

p̄2 (1) p̂2 (1) p̄2 (4) p̂2 (4)
,
,
,
p2 (1) p2 (1) p2 (4) p2 (4)

(bottom) with (in-

An inaccurate estimation of the support leads to an improper estimation of α0 and hence also
the probabilities of rare events. Under the current Example 4.2, we conduct a sensitivity analysis of
our estimation procedures by choosing a support set which is different from the one that is specified
by the model. In the example, the support set is identified by (a∗l , a∗u ) = (0.67, 1.5). Recall that we
have 30,000 data points from this model.
First we estimate α0 under an improper specification of al and au . This is estimated q
using a Hill
p
2
plot of points comprised of (X2 − au X1 )/ 1 + au for X2 − au X1 > 0 and (al X1 − X2 )/ 1 + a2l for
al X1 − au X2 > 0 for different choices of (al , au ). In Figure 9, we provide Hill plots for estimating α0
by using (al , au ) = (0.5, 2), (al , au ) = (0.9, 1.11), (al , au ) = (1, 1) respectively. Comparing the plots
with the one in Figure 6, the estimates clearly move away from the actual value α0 = 2.5 as the size
of the support decreases. When we take (al , au ) = (0.5, 2) the data points used to estimate α0 are a
subset of the points used to estimate α0 when (a∗l , a∗u ) = (0.67, 1.5), and thus are regularly varying
with parameter α0 = 2.5; hence the Hill plot clearly hugs the horizontal line at y = 2.5. As the size
of the support set decreases we see the Hill estimates become lower than 2.5 and moves towards
α = 1.5. We also observe in Figure 10 that the Hillish plots are not that close to the horizontal
line at height 1 when the support is not correctly identified; in this case (al , au ) = (0.9, 1.11).
In comparison, Figure 7 clearly shows that the Hillish plots are close to 1, when the support is
correctly specified.
Finally we estimate probabilities p1 (x) = P(X2 −2X1 > x) and p2 (x) = P(X2 −3X1 > x) when the
p̂1 (1) p̄4 (4) p̂1 (4)
support sets are identified incorrectly. Figure 11 corresponds to boxplots of p̄p11 (1)
(1) , p1 (1) , p1 (1) , p1 (4)
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

22

Das and Resnick
p̄2 (1) p̂2 (1) p̄2 (4) p̂2 (4)
p2 (1) , p2 (1) , p2 (4) , p2 (4)

where p̂i , i = 1, 2 uses the estimators given in (4.5),(4.6) with α0 comand
puted using (al , au ) = (0.5, 2) and p̄i , i = 1, 2 uses α0 = 2.5, al = 0.5, au = 2. Observe that the
boxplots are quite close to 1, since α0 is estimated well. On the other hand in Figure 12, the similar
boxplots are done for (al , au ) = (0.9, 1.11). In this case, α0 is not that well-estimated and hence
the boxplots are clearly away from 1. In both cases, 100 replications of data sets with 10,000 data
points in each were used to create the boxplots.
In conclusion we can see that an incorrect identification of the support of regular variation can
often lead to incorrect estimates. Although if the identified support of hidden regular variation is
a bit smaller than the correct one (which means that the identified support of MRV is larger than
the correct one), then the estimates are still quite accurate.

imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

23

5. Examples of strong asymptotic dependence with real data.
We now analyze two real data sets: (i) facebook wall posts and (ii) returns from Exxon and Chevron.

5.1. Facebook wall posts

1400

(out,in)−Facebook
●

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in degree

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Figure 13. Scatter plot of node-wise out-degree and in-degree of Facebook wallpost graph.

The Facebook wall posts data was downloaded from http://konect.uni-koblenz.de/networks/
facebook-wosn-wall and has been analyzed in [32]. Conversion of edge data to node-indexed inand out-degree counts was done using the R-package igraph [6]. The data is a directed network
representing posts by Facebook users to other users’ walls. Nodes are users and a directed edge represents one post from the user to the user whose wall is receiving the post. There are 46,952 users and
876,993 edges. We focus on out- and in-degree indexed by the nodes as {(Z1,i , Z2,i ) : 1 6 i 6 46952}.
Of course this data is not the result of iid replication but is rather node-indexed; however, for
reasons still being investigated, conventional tools of heavy tail analysis seem quite effective on
node-indexed network data. The scatter-plot of (out,in)-degrees in Figure 13 shows the expected
strong asymptotic dependence between out- and in-degrees.
The plots in Figure 14 give the estimation of distribution tail indices for out- and in-degree.
The slope estimator based on QQ-plots ([2; 22; 25]) gives approximately α = 2.8 for both out- and
in-degree. Note this estimate is for the tail of the cumulative distribution functions and not, as is
customary in network science, the index of the power law of the mass functions. The Hill estimator
is ineffective and we have provided altHill plots ([13; 25; 29]).
To get more information about the dependence structure, we construct a diamond plot using
thresholding corresponding to the 200 largest L1 norms of (out,in). The scatter plot in Figure 13
is less clear than for simulated data and shows points dispersed from the main cluster about the
diagonal and so the estimates of the support of θ in the diamond plot are not as evident as in
Figure 15.
imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

24

Das and Resnick
^ = 2.74
Facebook(in−degree): α
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Figure 14. (Left) Marginal estimation of tail indices by QQ plot slope estimation and (right) altHill plotting for
out- and in-degree. Horizontal lines are at height 2.8.
Histogram of θ1

Box plot of θ1

7

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Figure 15. Diamond plot restricted to the first quadrant for empirical angles thresholded using the 200 largest L1
norms, along with the histogram and boxplot.

We estimate the support interval using the interquartile range and obtain [.4479, .5305] = [θl , θu ].
This corresponds to slopes (al , au ) = (θu−1 − 1, θl−1 − 1) = (.885, 1.23). We also include a boxplot of
the values of θ1 corresponding to the 200 largest values of the L1 norm of (out,in).
Having determined [wedge], we remove it from the first
quadrant and use the remaining points, illustrated in Figure
16, to seek hidden regular variation. Preliminary diagnostics
use equations (3.6) and (3.7) corresponding to points above
and below [wedge] to estimate α0 , the index of hidden regular
variation. There are 12,089 points above [wedge] in the region
we refer to as [> wedge] = {(x, y) : y > (1.23)x > 0} and
24,687 below in the region [< wedge] = {(x, y) : 0 < y <
(0.885)x}.
For the region [< wedge], a combination of QQ and altHill
plotting gives a stable region for various values of k, the number of upper order statistics, between 100-500 and a value of
α̂0 = 2.8, the tail index of (0.885)Z1 − Z2 from (3.7), which is
Figure 16. Points remaining after re- not measurably different from α̂ = 2.8 found for the marginal
moval of [wedge].
distributions of (out,in). This raises doubts about the presence
of HRV in the region [< wedge]. For the region [> wedge] we
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imsart-bj ver. 2011/11/15 file: SDep_arxiv_2017.tex date: February 2, 2017

Strong Asymptotic Dependence

25
^ 0 = 3.19
QQ estimation: α
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●●●●

6

7

quantiles of exponential

8

9

0.3

0.4

0.5

0.6

0.7

θ

Figure 17. Facebook above [wedge]: QQ estimate is 3.2 for k = 100 and altHill is 3.3; the red line is at height 3.3.

use as data (Z2 − (1.23)Z1 )+ and estimate α̂0 ≈ 3.2. The QQ-estimate is 3.2 and altHill gives about
3.3; both estimates are greater than α = 2.8 so there is evidence of the existence of HRV in the
region above [wedge]. The QQ and altHill plots for the region [> wedge] are given in Figure 17. The
evidence for HRV is further strengthened by excellent Hillish plots described in Section 3.6 applied
to the generalized polar coordinates (Z2 − (1.23)Z1 , Z2 /Z1 ) as descibed in (3.8). Both Hillish plots
in Figure 18 hug the horizontal line at height 1.

Figure 18. Hillish plots for generalized polar coordinates of points in [> wedge].

The diamond plot in Figure 15 shows the presence of points on the line x1 + x2 = 1 with small
values of θ, corresponding to two dimensional points in [< wedge]. Previously the estimation of the
range of the angular measure of the primary regular variation discounted these points. However, the
estimation of the tail index of the distance to [wedge] being 2.8, the same as the marginal distribution
indices of (out,in), suggests an alternate model which lumps together [wedge] ∪ [< wedge] as the
region of concentration for the limit measure ν(·) of the primary regular variation in (2.4). So our
alternate model is regular variation on R2+ \{0} with index 2.8 and limit measure which concentrates
on {x ∈ R2+ \ {0} : x2 /x1 < 1.23} and hidden regular variation on R2+ \ ([wedge] ∪ [< wedge]) with
index 3.3.
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Das and Resnick

5.2. Exxon and Chevron returns.
XOM & CVX returns

CVX

0.00

60
0

1000

2000

3000

4000

−0.15

20

40

Price

80

0.10

100

XOM (red) & CVX (blue) price

−0.15

−0.10

Date

−0.05

0.00

0.05

0.10

0.15

XOM

Figure 19. Stock prices and scatterplot of Chevron and Exxon returns.

40
30
20

θ2

Frequency

10
0

−1.0

0.0

0.5

1.0

For this example of financial returns, the state space is R2 \ {0} and for the HRV property
we could try deleting [wedge+ ] ⊂ R2+ in the first quadrant and [wedge− ] ⊂ (−∞, 0)2 in the third
quadrant. For illustration, we concentrate on deleting only a wedge from the first quadrant and
seeking HRV with points above the upper boundary of [wedge+ ]. This is done partly because there is
no guarantee that HRV will hold globally. The data consists of closing daily prices of Exxon (XOM)
and Chevron (CVX) from January 2, 1998 to August 9, 2013. For each variable we calculate daily
returns for each company called (exxonr, chevronr). The length of the return vector is 3925. One
expects strong dependence from two big companies engaged in similar economic activities and this
is shown in the raw scatter plot of the variables in Figure 19.
The four tails of the variables (±exxonr, ±chevronr) are quite similar. Based on analyses (not
shown) using the QQ estimator, Hill and altHill plots, (eg. [22], [25, p. 101, 366]) we estimate
the marginal tail indices α = 2.7 in all four cases.
Since the tails are estimated to have the same α,
Histogram of θ1
Diamond plot; L1
we did not attempt to standardize the variables to
α = 1 as is often done by either the power method
or the ranks transform.
To understand the dependence structure of the
variables (exxonr,chevronr), we make a diamond plot
of the data. We do the mapping after thresholding
the data at various values determined by k, the num−1.0
0.0 0.5 1.0
−1.0
0.0 0.5 1.0
θ1
θ1
ber of order statistics of the norms |x1 | + |x2 |. This is
the two-tail empirical equivalent to (3.4) using the L1
Figure 20. Diamond plot for 200 largest values un- norm. After experimenting with thresholds, we setder L1 norm for (exxonr,chevronr) with histogram.
tled on k = 200 which in the first quadrant produced
a range of θ1 = x1 /(x1 + x2 ) equal to (.095, .649).
We finalized our estimate of the support of the limit angular measure, by using the 10% and 90%
quantiles of the values of θ1 as (.393, .589). This corresponds to slope estimates for [wedge] of
(âl , âu ) = (.698, 1.545). The strong asymptotic dependence among the marginals is evident from
the diamond plot and histogram of θ1 in Figure (20). There is little evidence that a large positive
change in one variable is accompanied by a large negative change in the other as shown by the lack
of points in the second and fourth quadrants in Figure 20. There is no visual evidence supporting
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Strong Asymptotic Dependence

27

the hypothesis of full asymptotic dependence.
Remark 3.1 suggests verifying the necessary
condition for HRV on R2 \ [wedge+ ] by computing the tail index of what is essentially the
distance of a point to [wedge+ ]. We seek evidence of regular variation on R2 \ [wedge+ ] by
using points x of the return sample that satisfy,

CVX

0.00 0.02 0.04 0.06

Returns XOM & CVX >[wedge]

1. x2 > 0 (points above the horizontal axis);
2. x2 − (1.545)x1 > 0 (points in the first
−0.06 −0.04 −0.02 0.00
0.02
0.04
0.06
or second quadrant above the ray x2 =
XOM
1.545x1 , x1 > 0)
3. x1 + 1.545x2 > 0 (points in the first or
Figure 21. Points satisfying three conditions allowing
second quadrant in the region bounded by computation of gpolar coordinates. Scanning counterclockthe ray x2 = 1.545x1 , x1 > 0 and the ray wise, the first two rays define [wedge+ ] and the third ray is
perpendicular to this ray emanating from perpendicular to the upper boundary of [wedge+ ].
the origin into the third quadrant). Points
to the left of this perpendicular would be
closer to [wedge− ] rather than [wedge+ ] and are excluded.
There are 706 points satisfying the three conditions; we call these points oilReturns2013Gr . These
are plotted in Figure 21. The angle between the two rays of biggest slope is 90 degrees.
We estimate the tail index α0 of the distance of points in oilReturns2013Gr to the boundary of
[wedge+ ] to be greater than α = 2.7 using altHill, Hill and QQ plots. This corresponds to estimating
the tail index of Z2 − au Z1 as in (3.6). The plots are given next.
α0 = 3.2, k=85

0.2

0.4
theta

0.6

0.8

−2.5
−3.0

** **
***
********
******
*
***
*****
*
****
*******
*****
*****

−4.0

0

0.0

*

−3.5

log sorted data

5
4
3
2
1

3.0

4.0

Hill estimate of alpha

5.0

Hill plot

2.0

Hill estimate of alpha

altHill

0

100

200

300

400

500

number of order statistics

600

700

2

3

4

** * * *

* * * *

5

*

6

quantiles of exponential

Figure 22. AltHill, Hill and QQ plots to estimate α0 . The red horizontal lines are drawn at height α = 2.7.

More evidence for existence of HRV in the region corresponding to oilReturns2013Gr is provided
by converting the data points in this region using the generalized polar coordinates suggested by
(3.8). This produces the Hillish plots given in Figure 23. Both Figures 22 and 23 are encouraging.

6. Conclusions
Whenever the limit measure of multivariate regular variation concentrates on a cone smaller than
the full state space, there is the potential for seeking hidden regular variation. This idea has been
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28

Hillish(R,−theta): data

Das and Resnick

0.8
0.4

Hillish statistic

0.8
0.4
0.0

Hillish statistic

1.2

Hillish(R,theta): data

0

200

400

600

number of order statistics

0

200

400

600

number of order statistics

Figure 23. Hillish plots for generalized polar coordinates of points in oilReturns2013Gr .

most often applied to the case of asymptotic independence where the limit measure concentrates
on the axes. Here we have shown the idea is also applicable when the limit measure concentrates
on the diagonal or a narrow cone such as [wedge].
Without hidden regular variation, asymptotic independence causes analysts to miss risk contagion. Analogously, when the limit measure concentrates on the diagonal, analysis would estimate
the probability of a risk region {(x, y) : y − x > 4} to be zero when in fact, hidden regular variation
would yield a small but non-zero probability. Our data analyses show potential for such estimation
in strongly dependent data with heavy-tailed marginal distributions.
Without doubt, much work remains to be done on implementation. Both our network data which
is node based and our returns data is nothing like independent replicated data. Also, our methods
for estimating the support of the angular measure S(·) are primitive at best. Higher dimensional
examples present increased visualization and estimation difficulties. None-the-less, we believe the
worked out examples are useful and illustrate practical cases. Other examples exist and in particular
we have analyzed Microsoft vs Dell returns with results similar to those found in Section 5.2.

7. Acknowledgements
We acknowledge with thanks the contribution in fall 2014 of Amy Willis who skilfully analyzed many
financial data sets seeking examples of asymptotic full and strong asymptotic dependence. We are
also grateful to Paul Embrechts who read an early draft and had many useful and encouraging
comments. Two referees made many helpful and insightful comments.
B. Das was supported by MOE-2013-T2-1-158 and IDG31300110. B. Das also acknowledges
hospitality from Cornell University during visits in June 2015 and January 2016. S. Resnick was
supported by Army MURI grant W911NF-12-1-0385 to Cornell University.

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