Multivariate Skewness: Measures, Properties and Applications

Transcription

1 Multivariate Skewness: Measures, Properties and Applications Nicola Loperfido Dipartimento di Economia, Società e Politica Facoltà di Economia Università di Urbino Carlo Bo via Saffi 42, Urbino (PU) ITALY nicola.loperfido@uniurb.it

2 Introduction Univariate skewness; Third moment; Mardia s skewness; Directional skewness.

3 Univariate skewness: First game s description Five friends bet one euro each as follows: Each friend puts a ticket with his signature in an urn; A ticket is randomly chosen from the urn; The friend who signed that ticket wins all the euros.

4 Univariate skewness: First game s payoff X 1 = E ( X ) = 0 V ( X ) = 4 1 1

5 Univariate skewness: Second game s description Five friends borrow an euro each, and agree to pay the money back as follows: Each friend puts a ticket with his signature in an urn; A ticket is randomly chosen from the urn; The friend who signed that ticket repays all the borrowed money.

6 Univariate skewness: Second game s payoff X 2 = E ( X ) = 0 V ( X ) = 4 2 2

7 Univariate skewness: Definition ( ) ( ) ( ) ( ) ( ) X X X E X X V X E X γ β σ γ σ = = = = R

8 Univariate skewness: Comparing the games Ε(X 1 )= Ε(X 2 )= 0 V(X 1 )= V(X 2 )= 4 γ 1 (X 1 )=1.5=-γ 1 (X 2 ) β 1 (X 1 )=β 1 (X 2 )=2.25

9 Univariate skewness: Interpretation When skewness is positive (negative), positive deviations from the mean contribute more (less) to variability than negative ones Values very different from the mean are often greater (smaller) than the mean itself

10 Univariate skewness: Symmetry Measures γ 1 and β 1 equal zero when the underlying distribution is symmetric X X γ ( X ) = β ( X ) = 0 1 1

11 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 5 2. Win 5 euros in the opposite case

12 Univariate skewness: Third game s payoffs X 3 = 4 2 / 6 1 3/ 6 5 1/ 6

13 Univariate skewness: Third game s properties The expected gain is zero P (win) = 2/3 > 1/3 = P (lose) Both γ 1 (X 3 ) and β 1 (X 3 ) equal zero

14 Problem How does skewness generalizes to multivariate distributions?

15 Kronecker product A = { } p q a R R B = { b } ij ij R r R s A B = { } p r q s a B R R i =1,..., p j = 1,..., q ij

16 Standardized random vector x R d E ( x) = V ( x) = Σ Σ 1/ 2 = ( 1/ 2 Σ ) T Σ 1/ 2 Σ 1/ 2 = Σ 1 Σ 1/ 2 > 0 z = Σ 1/ 2 ( x )

17 Third moment: Definition 3 ( ) ( ) 2 x = E x x T x R d R d

18 Third moment: Elements Structure Constraint Number E(X i3 ) none d E(X i2 X j ) i j d(d-1) E(X i X j X h ) i, j, h d(d-1)(d-2)/6 Total none d(d+1)(d+2)/6

19 Third moment: Example ( ) ( ) = = = = ,2,3,,, x h j i X X X E X X X x h j i ijh

20 Third moment: Linear transformations x R d A R k R d ( ) = ( A A) ( x) A T 3 Ax 3

21 Third moment: Symmetry The third moment of a centrally symmetric random vector is a null matrix ( x ) O d d x x = 3 2

22 Third moment: Sample counterpart m 3 ( X ) n 1 = n i=1 x i x T i x i X x i = = data matrix i-th column of X T

23 Third moment: Statistical applications Multivariate Edgeworth expansions; Skewness of financial data; Other measures of skewness.

24 Third moment: Essential references Franceschini, C., Loperfido, N., (2009). A Skewed GARCH-Type Model for Multivariate Financial Time Series. In Mathematical and Statistical Methods for Actuarial Sciences and Finance XII, Corazza M. and Pizzi C. (Eds.), ISBN: Kollo, T. (2008). Multivariate skewness and kurtosis measures with an application in ICA. Journal of Multivariate Analysis 99, Kollo, T. & von Rosen, D.(2005). Advanced Multivariate Statistics with Matrices. Springer, Dordrecht, The Netherlands

25 Mardia s skewness: Definition x,y R d E ( x) = μ V ( x) = Σ > 0 x,y are i.i.d. β M 1,d [ ] 3 1 ( x) = E ( x μ) Σ ( y μ)

26 Mardia s skewness: Invariance Mardia s skewness is invariant with respect to one-to-one affine transformations A R det d ( ) M ( ) M A 0 β x = β ( Ax + b) b R R d d 1, d 1, d

27 Mardia s skewness: Standardization Mardia s skewness is the squared (Frobenius) norm of the third standardized moment [ ] M 2, d ( ) ( ) T x = z tr ( z) ( z) β = 1

28 Mardia s skewness: Symmetry Mardia s skewness equals zero for centrally symmetric random vectors: x x β ( x) 0 M = 1, d

29 Mardia s skewness: Sample counterpart b 1, d 1 n n [ ] ( ) ( ) T 1 X = x m S ( x m) 2 i, j i j 3

30 Mardia s skewness: Statistical applications Multivariate normality testing; Assessing robustness in MANOVA; Parametric point estimation.

31 Mardia s skewness: Essential references Baringhaus, L. and Henze, N. (1992). Limit distributions of Mardia's measures of multivariate skewness. Ann. Statist. 20, Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika 57, Mardia, K.V. (1975). Assessment of Multinormality and the Robustness of Hotelling's T² Test. Appl. Statist. 24,

32 Directional skewness: Definition β ( ) ( ) x = max β c T x D 1,d d c R 1 0

33 Directional skewness: Invariance Directional skewness is invariant with respect to one-to-one affine transformations A R det d ( ) D ( ) D A 0 β x = β ( Ax + b) b R R d d 1, d 1, d

34 Directional skewness: Standardization Directional skewness is a function of the third standardized moment β ( ) [( ) ( ) ] T T x = c c z 2 D 1, d sup 3 c T c c= 1

35 Directional skewness: Sample counterpart ( ) , 1 max 0 = = R n i T T i T c d D S m x n X b d c c c c

36 Directional skewness: Statistical applications Multivariate normality testing; Projection pursuit; Parametric point estimation.

37 Directional skewness: Essential references Malkovich, J.F. and Afifi, A.A. (1973). On tests for multivariate normality. Journal of the American Statistical Association 68, Huber, P.J. (1985). Projection pursuit (with discussion). Ann. Statist. 13, Baringhaus, L. and Henze, N. (1991). Limit distributions for measures of multivariate skewness and kurtosis based on projections. J. Multiv. Anal. 38, 51-69

38 Hints for future research Interpretation Modelling Inference