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) ITALYnicola.loperfido@uniurb.it

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