elliptical distributions24
TRANSCRIPT
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Normal Distribution
Laplace Distribution t-Student Distribution Cauchy Distribution Logistic Distribution Symmetric Stable Laws
Examples of the Elliptical Distributions
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Examples of the Elliptical Distributions
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Examples of the Elliptical Distributions
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Bivariate NormalDistribution
1
1,0
0 ρ
ρ N
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The further from !ero the more e"ident
ellipticity of the map# when obser"ing itfrom abo"e$ %hen &' then the map hasthe spherical form$
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Laplace bivariatedistribution
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The random "ector is said toha"e an elliptical distribution withparameters "ector and the matri(
if its characteristic function can bee(pressed as
for some scalar function and whereand ) are gi"en by
Denition of ellipticaldistributions
T n X X X ),...,( 1=
)1( ×n µ )( nn×Σ
[ ] ),tt()μtexp()Xtexp( Σ⋅= T T T ii E ψ ψ
),...,,(t 21 nT
t t t = T
AA=Σ
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!haracteristic "unction of the#$mmetric #table Distributions
)texp()μtexp()t( α α σ φ
−⋅= i
)()μtexp()Xt( 2t ii E ψ ⋅⋅=⋅
[ ]
⋅−⋅= 2/
2texp)μtexp()t(α
α σ φ i
[ ]( )2/exp)( α α σ ψ ⋅⋅−=⋅
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*f + has an elliptical distribution# we write+ , , , where is called
characteristic generator of + and hence# thecharacteristic generator of the multi"ariatenormal is gi"en by
The random "ector + does not# in general#
possess a density but if it does# itwill ha"e the form
or some non-negati"e function calleddensity generator and for some constantcalled normali!ing constant$
),,( ψ µ Σn E ψ
).2/exp()( uu −=ψ
)x( X f
[ ])()()( 1 µ µ −Σ−Σ
= − x x g c
x f T nn
X
)(⋅n g nc
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+ , where is thedensity generator assuming thate(ists$
%lternative Denotin& of theElliptical Distributions
),,( nn g E Σ µ (.)n g (.)n g
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.(amples of the distributions that don/tha"e mean nor "ariance0
1ll stable distributions whose inde( ofstability is lower than 2# e$g$ Cauchy orLe"y$
Mean and !ovariance 'roperties
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Let + , # let 3 be a matri( and $ Then
,
Corollary$ Let + , $ Then ,
,
4ence marginal distributions of elliptical distributions are elliptical distributions$
Mean and !ovariance 'roperties
),,( nn g E Σ µ nq×q Rb∈),,( q
T
q g B B Bb E Σ+ µ BX b +
),,( nn g E Σ µ r X ),,( 111 r r g E Σ µ r n X − ),,( 222 r nr n g E −− Σ µ
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4ence follows that the sum of elliptical
distribution is an elliptical distribution$ Thisproperty is "ery important when we dealwith portfolio of assets# represented bysum$
!onvolutional 'roperties
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2$ .lliptical distributions can be seen as ane(tension of the Normal distribution
5$ 1ny linear combination of ellipticaldistributions is an elliptical distribution
6$ 7ero correlation of two normal "ariables
implies independence only for Normaldistribution$ This implication does not holdfor any other elliptical distribution$
Basic 'roperties of the EllipticalDistributions
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8$ + , with rank9):&k if + hasthe same distribution as
%here 9radius : and is uniformlydistributed on unit sphere surface in and
1 is a 9k;p: matri( such that
Basic 'roperties of the EllipticalDistributions
),,( φ µ Σ p E
)(k T u Ar ⋅+
µ 0≥r )(k u
n R
Σ=⋅ A AT
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!onstruction of a densit$function with innite variance
.2/if
%on$t
)(2
11
1)(
eaninfiniteha$then"1,2/1(If .1if
%on$t
)(2
11
1)(
)(2
11
1)(
2
1)(
2/1if
)1()(,
)1(
1)(
22
2
122
2
12
2
1
2
12
11
1
0
2/1
0
1
0
12/11
>
∞
∞
∞
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The e(pected shortfall 9or tail conditionale(pectation: is de
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or the familiar normal distribution N(μ, ), with mean μ and "ariance # it was noticed
by =an>er 95''5: that0
Expected #hortfall
2σ 2σ
2
1
1
)( σ
σ
µ
σ µ ϕ
σ µ
−Φ−
−
+=q
q
q X x
x
xTCE
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Suppose that g9(: is a non-negati"e functionfor any positi"e number# satisfying thecondition that0
Then g9(: can be a density generator of a
uni"ariate elliptical distribution of a random"ariable + ,
(enerali)ation of the'revious "ormula
∞
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The density of this function has the form0
where c is a normali!ing constant$ *f + has an elliptical distribution then
4as a standard elliptical distribution9spherical:
(enerali)ation of the "ormulafor the Normal Law
−=
2
2
1)(
σ
µ
σ
x g
c x f X
σ
µ −= X
!
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The distribution function of 7 has the form0
%ith mean ' and "ariance e?ual to
(enerali)ation of the "ormulafor the Normal Law
( ) ,duu2/1)( 2∫ ∞−
⋅= "
! g cu
).0()2/1(2 -
0
22ψ σ =⋅= ∫
∞
duu g uc !
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De
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Let + , and @ be the cumulati"egenerator$ Ander condition 9B:# the tailconditional e(pectation of + is gi"en by
%here is e(pressed as
*heorem +
),,( 2 g E n σ µ
,)( 2σ λ µ ⋅+=q X xTCE
)(2
1
)(2
1 22
q !
q
q X
q
"
" #
x
" #
=
=λ
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2$ or Cauchy distribution the TC. doesn/te(ist$ 3ecause it doesn/t satisfy conditionsof the theorem
Examples
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Lo&istic Distribution
( )
+
+
=
+=
+=
+=
=
+−=
−
−
−−
2tanh1
)(
2
1
)(1
2tanh
21
21)(
)(2
1
)(1
2
1
2
1
2
12
1
2
1
112
1
2
101
2
2
2
)2/1(
)2/1(
1)2/1(2
q
q
q
q
q
q !
q
q
"
"
"
q
"
"
"
TCE
" "
"
"
e
e
e " #
q
q
q
ϕ
π
ϕ σ
ϕ π
ϕ σ
π π
π
σ
σ σ
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Suppose + , is the"ector of ones with dimension n$ De
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The TC. can be e(pressed as
This theorem holds as a result ofcon"olution properties of the ellipticaldistributions and the pre"ious theorem$
*heorem
S S qq s
q s !
q s
S S
n
k $ k $
T
S
n
k k s
S S sqS
s "
"
" #
sTCE
σ µ
σ λ
σ σ µ µ µ
σ λ µ
/)(&ith
,)(
2
11
and
ee,e&here
)(
,
,
2
,
1,1 ,
2
1
T
2
−=
=
=Σ===
⋅+=
∑∑ ===
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Suppose + , is the"ector of ones with dimension n# and
Then the contribution of to theo"erall risk can be e(pressed as0
#ums of Elliptical ,is-s
T
nn g E )1,...,1,1(eand),,( =Σ µ
Xe...1
21
T n
k
k nn X X X X S ==+++= ∑=nk X k ≤≤1,
S k
S k
S k
S k S k S k qS X
nk
sTCE k
σ σ
σ ρ
ρ σ σ λ µ
,
,
,
&here
,...,2,1for
,)(
=
=⋅+=
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Skewed
#tableDistributions
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2$ *%/L !ND/*/N%L E1'E!*%*/N#", ELL/'*/!%L
D/#*,/B2*/N# 7ino"iy $ LandsmanB and .miliano 1$
Valde!E 5$ C1= and Option =ricing with .lliptical
Distributions# 4amada # Valde!$
6$ 4andbook of 4ea"y Tailed Distributions ininance# .ds S$T$ Fache"
Literatura