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Multivariate extremality measure

25 pages

We propose a new multivariate order based on a concept that we will call extremality". Given a unit vector, the extremality allows to measure the "farness" of a point with respect to a data cloud or to a distribution in the vector direction. We establish the most relevant properties of this measure and provide the theoretical basis for its nonparametric estimation. We include two applications in Finance: a multivariate Value at Risk (VaR) with level sets constructed through extremality and a portfolio selection strategy based on the order induced by extremality.
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Working Paper 10-19 Departamento de Estadística
Statistics and Econometrics Series 08 Universidad Carlos III de Madrid
June 2010 Calle Madrid, 126
28903 Getafe (Spain)
Fax (34) 91 624-98-49

Henry Laniado* Rosa E. Lillo** Juan Romo**


We propose a new multivariate order based on a concept that we will call
extremality". Given a unit vector, the extremality allows to measure the
"farness" of a point with respect to a data cloud or to a distribution in the
vector direction. We establish the most relevant properties of this measure
and provide the theoretical basis for its nonparametric estimation. We
include two applications in Finance: a multivariate Value at Risk (VaR)
with level sets constructed through extremality and a portfolio selection
strategy based on the order induced by extremality.

Keywords: extremality; oriented cone; value at risk; portfolio selection

* Department of Statistics, Universidad Carlos III de Madrid, 28911,
Leganés, Madrid, Spain. e-mail: hlaniado@est-econ.uc3m.es .

** Department of Statistics, Universidad Carlos III de Madrid, 28903,
Getafe, Madrid, Spain. e-mail: lillo@est-econ.uc3m.es (Rosa E. Lillo)
romo@est-ecom.uc3m.es (Juan Romo). Multivariate Extremality Measure
∗ †Henry Laniado Rosa E. Lillo Juan Romo
June 16, 2010
We propose a new multivariate order based on a concept that we will call
”extremality”. Given a unit vector, the extremality allows to measure the ”far-
nness” of a point in ℜ with respect to a data cloud or to a distribution in the
vector direction. We establish the most relevant properties of this measure and
provide the theoretical basis for its nonparametric estimation. We include two
applications in Finance: a multivariate Value at Risk (VaR) with level sets con-
structed through extremality and a portfolio selection strategy based on the
order induced by extremality.
1 Introduction
A multivariate order is a valuable tool to analyze the data properties and to obtain
direct analogues for multivariate data of univariate order concepts such as median,
range, extremes, quantiles or order statistics. Generalization of these concepts to
nthe multivariate case is not easy due to the difficulty of defining total orders in R .
but the lack of a unique criterion for ordering multivariate observations is the key
problem in extending these concepts to several dimensions. Barnett [3] was among
the first to give an extension of univariate order concepts such as median, extremes
and ranges to the higher dimensional case. A flexible way to summarize properties
of multivariate data are processes based on generalized quantile functions which are
studied in Einmahl and Mason [11].
Multivariate orders allow comparisons and decision making in multiple output
scenarios; for example, in psychology and sociology to compare individuals by their
characteristics; in the financial industry is important to compare portfolios and per-
formance of investment funds(Zani et al [29]). The detection of outliers in multivari-
ate data is also a relevant application of the multivariate orders (Cerioli and Riani
nSeveral extensions of usual orders from R to R , such as the Pareto-dominance
types and componentwise order, have the drawback of not being total orders. To
∗Department of Statistics, Universidad Carlos III de Madrid, 28911, Legan´es, Madrid, Spain
(e-mail: hlaniado@est-econ.uc3m.es)
†Department of Statistics, Universidad Carlos III de Madrid, 28903, Getafe, Madrid, Spain (e-
mail: lillo@est-econ.uc3m.es; juan.romo@uc3m.es)facilitate the total comparison in the multivariate case the antisymmetry is waived;
as a consequence, preorders are obtained instead of orders. An example is defining
nsome function of interest f : R −→ R and ordering the data according to its f-
value i.e., x ≤ y ⇐⇒ f(x) ≤ f(y). Orders defined through either norms or pro-
jections onto some vector ~u such as order by average or weighted average are of
nthis type (see Barnett [3]). A depth function assigns each point in R a measure of
centrality with respect to the data cloud or probability distribution. This measure
decreases from the center outward (see, e. g., Zuo and Serfling [30] and Liu et al.
[19]) and thus a depth function provides a multivariate order that allows to define
multivariate versions of median, order statistics, multivariate spacing and tolerance
regions(LiandLiu[20]). Anotherexampleismajorization (MarshallandOlkin[22]).
The majorization order is based on the idea of homogeneity between the components
nof a vector in R and is used in economy to compare the distribution of wealth in
populations. Other orders can be characterized by a Euclidean convex cone C; for
ninstance, for x, y∈R x≤y⇐⇒y−x∈C. This is the case of the componentwise
+ +order, where C =R ∪{0} or C =R . These are two of the most important convex
cones: the non-negative orthant and the positive orthant and are useful in the theory
of inequalities. It is customary to write x≤ y, if y−x belongs to the non-negative
orthant. (see Rockafellar [24]).
Next, we introduce the concept of extremality for multivariate data. The extre-
nmality of x∈R in the direction ~u is one minus the probability of a oriented convex
cone with vertex in x. This cone will be called oriented sub-orthant. Different ~u
unit vectors define different ways to rank a multivariate sample. An important step
in multivariate data analysis taking into account directions has been made in Kong
and Mizera [18] where is adopted a very simple and quite natural projection- based
definition of quantiles. More recently Hallin et al [15] proposed a new multivariate
quantile based on a directional version of traditional regression quantiles, which also
are associated with a vector ~u. In the same paper they showed that the contours
generated by the directional quantiles coincide with the classical halfspace depth
contours. Our proposal of extremality is also based on directions. To calculate the
nextremality of a point x ∈ R we move the non-negative orthant in the direction
given by~u and translating the origin to x. The above determines a isomorph cone to
u~the non-negative orthant and will be denoted by C . Thus, the extremality of x inx
u~direction ~u will be 1−P C . Unlike Hallin et al [15] where~u indicates the directionx
of the ”vertical” axis in the regression, in this paper ~u is ”bisectrix” of the oriented
1 u~ n√cone. For example, if ~u = 1 then,C = x+R . As a consequence of the extre-n x +n
mality concept we propose a order for multivariate data that allows to establish the
n”farness” of x∈R respect to points cloud or to a distribution function. Thus, this
extremality measureprovidesastatistical methodology forsegmenting amultivariate
∗ n u~data sample, since the set of x ∈ R such that P C = q can be interpreted as∗x
1 ′√a multidimensional quantile in the sense of Tibiletti [27] when ~u = [±1,...,±1].
In fact, extremality is a starting point to study segmentation by considering other
type of directions such as the first principal component. The purpose of this paper
is to analyze structural properties of this multivariate order and to initiate a theory
of nonparametric statistical estimation of the extremality in the sample case. We
introduce an estimator and prove its weak and strong consistency.
From an applied perspective to insurance and finance, first we propose a version
2of multivariate Value at Risk (VaR) based on extremality. The VaR as risk measure
has taken place as benchmark in the risk management techniques. Its approach
is based on a general notion of risk as the probability of not exceeding a certain
threshold quantity considered as dangerous. It has been strongly criticized from
Artzner [2] for not to encourage the diversification and defended by Heyde et al. [16]
for the robustness. For univariate risks, the VaR is simplytheα− quantile of the loss
the most popular measure used by risk managers. However, for the multivariate
case to define VaR is more complicated due to existence of manyfold definitions of
multidimensional quantile (see Einmahl and Mason [11], Tibiletti [27], Chaudhuri
[7], Serfling [25] and Hallin et al. [15] for definitions of multidimensional quantile).
Bivariate versions of VaR have been studied in Arbia [1], Tibiletti [28], Nappo and
more recently in Cascos and Molchanov [5]. We propose a multivariate VaR based
on extremality notion as its set levels. It enables to identify those relevant events for
1 ′√risk management in the direction ~u. Specifically ~u = [±1,±1] our VaR coincides
−1 ′ 1 ′√ √with the VaR in sense Tibiletti [28]. Now,~u= [1,...,1] and~u= [1,...,1] the
n n
VaR in this paper coincides with multivariate lower orthant VaR and multivariate
upper orthant VaR respectively, discussed in Embrechts and Pucceti [12]. However,
taking into account other directions we can obtain conservative types of VaR.
As a second application of the multivariate order based on extremality we pro-
pose a portfolio selection strategy. Portfolio selection problem was considered in
Markowitz [21] whose philosophy is that a investor should hold a portfolio on the
set of couples risk-return which one cannot improve both at the same time. This set
was denoted as the efficient frontier. From Markowitz [21], several criteria have been
studied (see for instance, DeMiguel et al. [10] and references there in) for portfolio
selection. We propose to sort feasible portfolios according to the order induced by
1the direction ~u = [1,−1] that favors the risk and does not favors the returns so we
must select the smallest portfolio. We also show that the portfolio selected under
this strategy belongs to the efficient frontier.
Thepaperisorganizedasfollows. Section2introducesthedefinitionandproperties
of the oriented sub-orthant and how it is constructed. In Section 3, we present the
extremality measure and the induced multivariate order. The main properties and
consistency results are discussed in Section 4. A multivariate VaR is proposed in
Section 5 and a portfolio selection strategy is constructed in Section 6, where we
compare it with strategies previously used in the literature. Finally, in Section 7 we
summarize the main conclusions.
2 Preliminaries
We introduce in this section definitions and preliminary results needed throughout
the paper. Recall that a binary relation on an arbitrary set C is called a partial
order if it satisfies: reflexivity (x x, for all x ∈ C), transitivity (x y and
y z =⇒ x z) and antisymmetry (x y and y x =⇒ x = y). Orderings that
nsatisfy reflexivity and transitivity are called preorders. A subset C of R is said to
be convex if (1−λ)x+λy∈C whenever x∈C,y∈C and 0≤λ≤1.
3nDefinition 1 (Extreme Point) Let C ⊆ R be a convex set. Then x ∈ C is an
extreme point of C if there not exist x ,x ∈C, x =x , such that1 2 1 2
x=λx +(1−λ)x , for some 0<λ<1.1 2
An extreme point of a convex set C does not belong to the segment between any
two points in C. Points 1 and 3 in the left panel of Figure 1 are extreme, while the
right panel has a unique extreme point.
Figure 1: Points 1 and 3 are extreme, and 2 and 4 are not extreme points
nDefinition 2 AsubsetC ofR isaconewithvertexinv ifv+λ(x−v)∈C, for allx∈
C andλ>0.
C is a convex cone if it is a convex set and satisfies Definition 2. Clearly the nonneg-
ative upperorthantisa convex cone with vertex in0. Inthispaper,we areinterested
in rotations of this cone. To formalize the idea, we will use the QR factorization.
Definition 3 Let A be a m×n matrix with m≥n. Then A can be factorized as
A =QR,
where Q is an orthogonal matrix and

R1R = ,
with R an upper triangular matrix.1
Matrix Q can be obtained by using, for instance, Householder Reflections, Givens
Rotations or Gram-Schmidt Transformations (see Gentle [14], pages 95-103). Since
′ −1Q is an orthogonal matrix, Q =Q . If the diagonal entries of R are required to be
nonnegative, Q and R are unique (we will assume nonnegative elements in R along
the paper). The next result establishes that R is the first element of the canonical
nbasis inR in the QR factorization of any unit vector.
′Proposition 1 Let ~u = [u ,...,u ] be a vector with Euclidean norm k~uk = 1. If1 n 2
′~u=QR, then R =[1,0,...,0].
Proof. We have that
′ ′ ′ ′ ′1 =~u~u=RQQR =RIR =RR.
′Therefore, R has to be [1,0,...,0] according to Definition 3.
61 ′ n√Consider the unit vectors ~e = [1,...,1] and ~u∈R . Writing
~e =Q R and ~u=Q R ,1 1 2 2
′R =R =[1,0,...,0] from Proposition 12 1
′ ′ ′. Hence, Q ~u=Q ~e and Q Q ~u=~e. Thus,12 1 2
′R =Q Q (1)1u~ 2
is an orthogonal matrix transforming~u into a unit vector with identical components.
This transformation will send each vector x to a new orthogonal coordinates system,
where~uhasall its angles equal with respectto the newnonnegative axis coordinates,
that is, R ~u = ~e. This transformation (1) allows to define the following specialu~
oriented cones.
u~ nDefinition 4 (Oriented sub-orthant C ) Givenaunitdirector vector~u∈R andx
n u~a vertex x∈R , an oriented sub-orthant C is the convex cone given byx
u~ nC ={z∈R |R (z−x)≥0}, (2)u~x
where the inequality is componentwise.
u~C is a convex cone with vertex in x obtained moving the nonnegative orthant andx
u~translating the origin tox. Besides, according to Definition 1,C is a convex set withx
a single extreme point in x, the semi-line
nl ={z∈R |z =x+λ~u, λ≥0} (3)
u~istotallycontained inC anditsangleswithrespecttothenewnonnegativesemi-axisx
−1 1 1 ′√ √coordinates are equal to cos . Note that when ~u= [±1···±1] and v =0,
n n
u~ n nC coincides with the 2 orthants inR .v

x12 ′Example 1 Consider R . If with ~u=[u ,u ] and x = then,1 2 x2
( )√
z 2 u +u u −u z −x 0u~ 1 2 1 2 2 1 1 1C = ∈R : ≥ . (4)x z 2 u −u u +u z −x 02 1 2 1 2 2 2
2Example 2 In R , the director vector ~u can be determined by an angle 0≤ θ≤ 2π
′indicating the direction of the cone. Then ~u=[cosθ,sinθ] and

π πz cos(θ− ) sin(θ− ) z −x 01 1 1u~ 2 4 4C = ∈R : ≥ . (5)x π πz −sin(θ− ) cos(θ− ) z −x 02 2 24 4
u~Thus, C is a convex cone obtained rotating the non-negative quadrant by an anglex
π ′(θ− ) and translating the origin to (x ,x ). Besides, the semi-line (3) will be1 24
1u~ −1 √bisectrix ofC with angles cos with respect to the rotated nonnegative semi-x 2
u~ u~ u~ u~1 1 2 3Figure 2 presents the oriented sub-orthants C ,C ,C andC with vertices inA B C D
π π 5π πA,B,C,D and ~u=[cosθ,sinθ],for θ = , , , , respectively.
3 4 4 2
Figure 2: Examples of oriented sub-orthants
π 3π 5π 7π
4 4 4 4Clearly,C , C ,C andC are(+,+); (−,+); (−,−); (+,−)quadrants,respec-+ + + +0 0 0 0
u~ u~n ′ 1 2tively. InR , if ~u =~e, ~u =−~e thenC andC are, respectively, the nonnegative1 2 0 0
and nonpositive orthants, sinceR =I andR =−I (see equation (1)).n nu~ u~1 2
u~ u~ u~Proposition 2 For any ~u, if x∈C then C ⊂C .y x y
u~Proof. Suppose that z ∈C . From Definition 4, R (z−x)≥ 0, and R (x−y)≥ 0u~ u~x
u~byhypothesis. Then,R (z−y)=R (z−x)+R (x−y)≥0and,therefore,z∈C . u~ u~ u~ y
The following Proposition shows that there exists at least a transformation that
nallows to compare componentwise two points inR .
(x−y)nProposition 3 If x = y ∈ R and ~u = , where k·k is the Euclidean norm
i) R y≤R xu~ u~
u~ u~ii) C ⊂C .x y
Proof. i) According to transformation (1), for any unit vector ~u, R ~u = ~e. Inu~
(x−y)particular, for ~u = , then R (x−y)≥ 0 and therefore, R y ≤R x. ii) Sinceu~ u~ u~kx−yk
u~ u~ u~R (x−y)≥0, x∈C and from Proposition 2 C ⊂C . u~ y x y
3 Extremality Measure
nLet F be the class of distribution functions on R , let X be a random vector with
distribution function F ∈ F and probability distribution P . Given a unit vector ~uF
n u~and x∈R , denote by P the measure P ofC , that is,x,u~ F x
u~P = dP =P C . (6)F Fx,u~ x
6IfP isabsolutelycontinuousandthemultivariate randomvectorX hasjointdensityF
function f , thenX
P = f (x)dx. (7)x,u~ X
2Proposition4showsthegeneralwaytocalculateP whenx∈R foranyorientation.x,u~
Proposition 4 If (X,Y) is a random vector with joint density function f andXY
′~u=[cos(θ),sin(θ)] as in the Example 2. Then,
 π π πxcos(θ− )+ysin(θ− )−tcos(θ− )Z Z 4 4 4 ∞ πsin(θ− )
4u~  P C = f (t,s)dsdt 1 π 7πF XYπ π π(x,y) θ∈ 0, ∪ ,2π−xsin(θ− )+ycos(θ− )+tsin(θ− ) { [ ) ( )}4 44 4 4x π
cos(θ− )
4 π π π
xsin(θ− )−ycos(θ− )+scos(θ− )Z Z 4 4 4∞ πsin(θ− )
4 + f (t,s)dtds 1 π 3πXYπ π π θ∈ ,xcos(θ− )+ysin(θ− )−ssin(θ− ) { ( )}
4 4 4 4 4y πcos(θ− )
4 
π π π−xsin(θ− )+ycos(θ− )+tsin(θ− )Z Z 4 4 4∞ π
cos(θ− )4 + f (t,s)dsdt 1 3π 5ππ π π XYxcos(θ− )+ysin(θ− )−tcos(θ− ) {θ∈( , )}
4 4 4 4 4x πsin(θ− )
4 
π π πxcos(θ− )+ysin(θ− )−ssin(θ− )Z Z 4 4 4y πcos(θ− )
4 + f (t,s)dtds 1 5π 7ππ π π XY θ∈ ,xsin(θ− )−ycos(θ− )+scos(θ− ) { ( )}4 44 4 4−∞ π
sin(θ− )4
Z Z Z Z ∞ ∞ x ∞
+ f (t,s)dsdt 1 π + f (t,s)dsdt 1 3πXY XY{θ= } {θ= }
4 4x y −∞ y
Z Z Z Z x y ∞ y
+ f (t,s)dsdt 1 + f (t,s)dsdt 1 .5π 7πXY XYθ= θ={ } { }4 4−∞ −∞ x −∞

u~In higher dimensions is more difficult to give a general expression for P C unlessF x
the unit vector ~u is given numerically.
′Let t =R x be the image of x under the transformation (1). Clearly, x =R tu~ u~
nand the absolute value of the Jacobian is 1. If we write D ={t∈R |t≥x}, thenx
(7) is equivalent to
−1P = f (R t)dt. (8)x,u~ X u~
If x ,...,x is a sample of the random vector X, the empirical version of P is1 m x,u~
given by
m mX X1 1ˆP = 1 = 1 , (9)x,u~ ~u {R (x −x)≥0}x ∈C ~u j{ j }xm m
j=1 j=1
u~which is the proportion of the data cloud belonging toC .x
As we have shown, P is the probability of an oriented sub-orthant. We canx,u~
now formulate the extremality notion. This concept is the starting point for defining
a new multivariate data order.
7nDefinition 5 (Extremality Measure) The extremality of x ∈ R with respect to
n +a distribution function F in direction ~u is a mappingE (x,F) :R ×F−→R ∪{0},u~
defined by

u~E (x,F) =P C =1−P , (10)Fu~ x x,u~
where P is given by (6).x,u~
nThe extremality of x∈R respect to a data cloud X ={x ,...,x } in direction ~u,1 n
ˆdenoted byE (x,X), is defined replacing P by P .u~ x,u~ x,u~
u~High extremality of a point x means that the convex cone C contains a smallx
part of the total mass and possiblyx belongs to some tail of the distribution. Hence,
high extremality can be interpreted as ”farness” in the distribution.
Figure 3 presents the extremality curves of level 0.99; 0.95; 0.90; 0.85 when F
is a bivariate distribution with independent marginal distributions U(0 1). Left side
1√in direction ~u = [1, 1] and right side in direction [0, 1]. Figure 4 shows the
1√extremality surfaces of level 0.99; 0.95; 0.90; 0.85 in the direction ~u= [1, 1, 1] of
a multivariate distribution F with three independent marginal distributions U(0 1).
All points on a particular curve or surface have the same extremality with respect to
the distribution F.
Figure 3: E 1 (x,F) =α andE (x,F) =α[0, 1]√ [1, 1]
8Figure 4: E 1 (x,F) =α√ [1, 1, 1]
Definition 5 induces a multivariate order as follows.
nDefinition 6 Given x,y ∈R , y is said to be more extreme than x respect to F in
direction ~u, denoted x≤ y if, and only if,E~u
E (x,F)≤E (y,F).u~ u~
nFor any x,y ∈ R , any distribution function F ∈ F and any ~u it holds that either
nx≤ y or y≤ x. However,≤ is not a partial order in R , but a preorder. Be-E E E~u ~u ~u
cause, although it satisfies reflexivity and transitivity properties, it does not satisfy
antisymmetry. If F is an absolutely continuous distribution in the interval [a, b], the
extremality order with ~u=1 coincides with the usual order inR.
4 Properties of extremality measure
The extremality measures are nonnegative and bounded. Next we establish its ana-
lytic properties that support the ordering proposed in the previous definition.
nProperty 1 For any x ∈R and any absolutely continuous F ∈F,0
E (x ,F) is continuous in ~u.u~ 0
Proof. Let f be the density function corresponding to F. From (8) and DefinitionX
5, we have that Z
−1E (x ,F) =1− f (R t)dt,u~ 0 X u~
−1which clearly is continuous in ~u sinceR is a linear transformation.
Thefollowing propertyindicates that the vertexxhasminimalextremality in the
u~setC .x