Multivariate Skewness:Measures, Properties and Applications
Nicola LoperfidoDipartimento di Economia, Società e Politica
Facoltà di EconomiaUniversità di Urbino “Carlo Bo”
via Saffi 42, 61029 Urbino (PU) [email protected]
Introduction
Univariate skewness;
Third moment;
Mardia’s skewness;
Directional skewness.
Univariate skewness: First game’s description
Five friends bet one euro each as follows:
Each friend puts a ticket with his signaturein an urn;
A ticket is randomly chosen from the urn; The friend who signed that ticket wins all
the euros.
Univariate skewness: First game’s payoff
( ) ( ) 40
2.08.0
41
11
1
==
−=
XVXE
X
Univariate skewness: Second game’s description
Five friends borrow an euro each, and agreeto pay the money back as follows:
Each friend puts a ticket with hissignature in an urn;
A ticket is randomly chosen from the urn; The friend who signed that ticket repays all
the borrowed money.
Univariate skewness: Second game’s payoff
( ) ( ) 40
8.02.0
14
22
2
==
−=
XVXE
X
Univariate skewness: Definition
( ) ( )
( ) ( ) ( )XXXEX
XVXEX
211
3
1
2
γβσµγ
σµ
=
−
=
==ℜ∈
Univariate skewness:Comparing the games
Ε(X1)= Ε(X2)= 0
V(X1)= V(X2)= 4
γ1(X1)=1.5=-γ1(X2)
β1(X1)=β1(X2)=2.25
Univariate skewness:Interpretation
When skewness is positive (negative), positivedeviations from the mean contribute more
(less) to variability than negative ones
Values very different from the mean areoften greater (smaller) than the mean itself
Univariate skewness: Symmetry
Measures γ1 and β1 equal zero whenthe underlying distribution is symmetric
( ) ( ) 011 ==⇒−≡− XXXX βγµµ
Univariate skewness: Third game’s description
Cast a regular dice:
Win 1 euro if the outcome is smaller than 4
Cast it again if the outcome is greater than 3:1. Lose 4 euros is the second outcome is smaller than 52. Win 5 euros in the opposite case
Univariate skewness: Third game’s payoffs
−
=6/16/36/2
5143X
Univariate skewness: Third game’s properties
The expected gain is zero
P (win) = 2/3 > 1/3 = P (lose)
Both γ1(X3) and β1(X3) equal zero
Problem
How does skewness generalizesto multivariate distributions?
Kronecker product
{ } { }
{ } qjpiBaBA
bBaA
sqrpij
srij
qpij
,...,1,...,1 ==ℜ×ℜ∈=⊗
ℜ×ℜ∈=ℜ×ℜ∈=
⋅⋅
Standardized random vector
( ) ( )
( )
( )µ
µ
−Σ=
>ΣΣ=ΣΣΣ=Σ
Σ==ℜ∈
−
−−−−−−
xz
xVxEx
T
d
2/1
2/112/12/12/12/1 0
Third moment:Definition
( ) ( ) ddT xxxEx ℜ×ℜ∈⊗⊗=2
3µ
Third moment: Elements
Structure Constraint Number
E(Xi3) none d
E(Xi2Xj) i ≠ j d(d-1)
E(XiXjXh) ≠i, j, h d(d-1)(d-2)/6
Total none d(d+1)(d+2)/6
Third moment: Example
( ) ( )
=⇒==
=
333233133
233223123
133123113
233223123
223222122
123122112
133123113
123122112
113112111
3
3
2
1
3,2,1,,,
µµµµµµµµµµµµµµµµµµµµµµµµµµµ
µµ xhjiXXXEXXX
x hjiijh
Third moment: Linear transformations
( ) ( ) ( ) TAxAAAx 33 µµ ⊗=
dkd Ax ℜ×ℜ∈ℜ∈
Third moment: Symmetry
The third moment of a centrallysymmetric random vector is a null matrix
( )ddOxxx
×=−⇒−≡− 23 µµµµ
Third moment:Sample counterpart
( )
Ti
n
ii
Tii
mn of X i-th coluxix data matrX
xxxn
Xm
=
=
⊗⊗= ∑=1
31
Third moment: Statistical applications
Multivariate Edgeworth expansions;
Skewness of financial data;
Other measures of skewness.
Third moment: Essential references Franceschini, C., Loperfido, N., (2009). A Skewed
GARCH-Type Model for Multivariate Financial TimeSeries. In “Mathematical and Statistical Methods forActuarial Sciences and Finance XII”, Corazza M. andPizzi C. (Eds.), ISBN: 978-88-470-1480-0
Kollo, T. (2008). Multivariate skewness and kurtosismeasures with an application in ICA. Journal ofMultivariate Analysis 99, 2328-2338
Kollo, T. & von Rosen, D.(2005). Advanced MultivariateStatistics with Matrices. Springer, Dordrecht, TheNetherlands
Mardia’s skewness: Definition
( ) ( )
( ) ( ) ( )[ ]311
0
μyΣμxExβ
i.i.d.arex,yΣxVμxEx,y
M,d
d
−−=
>==ℜ∈
−
Mardia’s skewness: Invariance
Mardia’s skewness is invariant with respectto one-to-one affine transformations
( ) ( ) ( )bAxxb
AA
Md
Md
d
dd
+=⇒
ℜ∈≠ℜ×ℜ∈
,1,10det ββ
Mardia’s skewness: Standardization
Mardia’s skewness is the squared (Frobenius)norm of the third standardized moment
⇓( ) ( ) ( ) ( )[ ]zztrzx TM
d 332
3,1 µµµβ ==
Mardia’s skewness: Symmetry
Mardia’s skewness equals zero forcentrally symmetric random vectors:
( ) 0,1 =⇒−≡− xxx Mdβµµ
Mardia’s skewness: Sample counterpart
( ) ( ) ( )[ ]∑ −−= −n
jij
Tid mxSmx
nXb
,
312,1
1
Mardia’s skewness: Statistical applications
Multivariate normality testing;
Assessing robustness in MANOVA;
Parametric point estimation.
Mardia’s skewness: Essential references
Baringhaus, L. and Henze, N. (1992). Limitdistributions of Mardia's measures of multivariateskewness. Ann. Statist. 20, 1889-1902
Mardia, K.V. (1970). Measures of multivariateskewness and kurtosis with applications. Biometrika57, 519-530
Mardia, K.V. (1975). Assessment of Multinormality andthe Robustness of Hotelling's T² Test. Appl. Statist.24, 163-171
Directional skewness: Definition
( ) ( )xcx T
c
Dd d 1,1
0
max ββℜ∈
=
Directional skewness: Invariance
Directional skewness is invariant with respect to one-to-one affine transformations
( ) ( ) ( )bAxxb
AA
Dd
Dd
d
dd
+=⇒
ℜ∈≠ℜ×ℜ∈
,1,10det ββ
Directional skewness: Standardization
Directional skewness is a functionof the third standardized moment
( ) ( ) ( )[ ]231
,1 sup czccx TT
cc
Dd
Tµβ ⊗=
=
Directional skewness: Sample counterpart
( )23
1,1
1max0
−= ∑
=ℜ∈
n
iT
Ti
T
cd
D
Smx
nXb
d cccc
Directional skewness: Statistical applications
Multivariate normality testing;
Projection pursuit;
Parametric point estimation.
Directional skewness: Essential references
Malkovich, J.F. and Afifi, A.A. (1973). On tests formultivariate normality. Journal of the AmericanStatistical Association 68, 176-179
Huber, P.J. (1985). Projection pursuit (withdiscussion). Ann. Statist. 13, 435-475
Baringhaus, L. and Henze, N. (1991). Limitdistributions for measures of multivariate skewnessand kurtosis based on projections. J. Multiv. Anal. 38,51-69
Hints for future research
Interpretation
Modelling
Inference