Tilburg University

A useful fourth moment matrix of a random vector

Tigelaar, H.H.

Publication date:1993

Link to publication

Citation for published version (APA):Tigelaar, H. H. (1993). A useful fourth moment matrix of a random vector. (Research memorandum / TilburgUniversity, Department of Economics; Vol. FEW 589). Unknown Publisher.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

- Users may download and print one copy of any publication from the public portal for the purpose of private study or research - You may not further distribute the material or use it for any profit-making activity or commercial gain - You may freely distribute the URL identifying the publication in the public portal

Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 03. Feb. 2021

7626i983589

r ~~~ J~V ~~S

oip `~,~ ~,cc~~o~~~ooo~~.~~~.~~

Q h ~

i r rrrrr~~~

6qnIIIIIIIIIIIIIIIIIIII IIId IIIIIN~]IIII~I

rr `~ írírírií

l~e

E ~~~~~K.E) !~.i

~ r-~i~ ~p~4 -~~~THEEK; ~~~~ ~ s~~u~Ui

A USEFUL FOURTH MOMENT MATRIX OF ARANDOM VECTOR

Harry H. Tigelaar

FEW 589

Communicated by Prof.dr. B.B. van der Genugten

A liSEFUL FOURTH MOMENT MATRIX OF A RANDOM VEGTOR

Harry H. Tigelaar

Tilburg University Econometrics dept.

Postbox 90153

5000 LE Tilburg

The Netherlands

Abstract

Let x be a real random n-vector. Then the fourth moments can be put into asquare matrix of dimensions n2xn2 in a lot of different ways. In this paper wepropose a fourth moment matrix which has nice properties when calculationsinvolving fourth moments have to be carried out such as the correlationbetween two quadratic forms in x. Furtherrnore it is shown how to calculatethis matrix from the distribution of x by introducing a suitable differentialoperator acting on the characteristic function.

Keywords characteristic functions, vector differentiation, Kronecker product,permutation matrices.

1

1. Introductton

Let x-(xl,...,xn)T be a real random n-vector with characteristic function cpwhich is four times differentiable. Then then all fourth moments of x arefinite and are given by

E(xjx~x~xm) - Lau~áukáu; u,~J j,k,l,m e{1,...,n}

u~o

where u-(ul,...,uo)Te R". Clearly any arrangement of these moments into amatrix corresponds to an arrangement of fourth order partial derivatives ofthe characteristic function ~p. Therefore we consider first vectordifferentiation of arbitrary matrix valued functions.Let

tli : Cn. C`" and ~ : CnYQ. C xP

be differentiable funtions where Cnx4 represents the space of complex nxqmatrices. The usual way of putting their partial derivatives into a matrix is

a~~ a~(Z)z, zo

a1~1(2) . . a VCC~(Z)T - . . and D~L(Z) - T .

az a~m(z) a~ (z) a(„e~z)az, ...-~zo

According to Magnus and Neudecker (1988) this is the only good way to do thisand indeed, when differentials are to be calculated it is the only method thatmakes sense. However, in our case we are dealing with repeated differentiationwith respect to a vector rather than a matrix and we are merely interested inthe values at zero rather than in differentials. Furthermore we want anarrangement of fourth order partial derivatives in a matrix in such a way thatit corresponds to a matrix of fourth moments which is suitable for matrixmanipulations. A matrix containing all fourth moments with nice properties is

Q - E{XXT~XXT}

2

where ~ stands for the Kronecker product. Two well known and frequently usedproperties of Kronecker products are

( A~B ) ( C~D ) - ( AC)~(BD )

vec(AXC) - (CT~A)vec(X) .

We shall refer to the matrix Q as the Quadruple matrix. As an example df itsuse we give an expression for the product moment of two quadratic forms interms of Q

E(xTAx)(xTBx) - tr{(A~B)Q} .

Proof. The result follows immediately from the observation that

(xTAx)(xTBx) - tr((xTAx)~(xTBx)] - tr[(xT~xT)(A~B)(x~x)] -

- tr[(A~B)(xT~xT)(x~x)] O

In Lhe next sections we develop a differential operator that enables us tocalculate the quadruple matrix from the characteristic function and give anapplication.

2. Differential Operators

In this section we consider the class of complex matrix valued functions of avector ueCn which are component wise partially differentiable and define the

differential operators D~ and D~ by

D~~p(u) -

r acp(u)I a~,

and D~~p(u) - (D~~p(u)T)T .

a~p ( u )ÓUn

When the argument vector u does not play a role we shall shortly write D~cp.

3

The symbols D~ and D; are motivated by the fact that De~p(u) looks like the

Kronecker product of the operator vector (a~aul,...,a~aun)T and the matrix

~p(u). Clearly we have DeuT- In (the nxn unit matrix) and Deu - vecIn. The

operators turn out to be an adequate tool in particular when Kronecker

products are involved. In particular the Quadruple matrix can now be written

as

Q - [ DeD~.De~G(u)] ~:a .

Before we can derive the rules of calculation for the differential operatorswe have to introduce some notation and list some properties of permutationmatrices.

3. Permutation Matrices

Let e~eCr denote the i`h unit vector in Cr (i-1,...,r) and éi the

vector in CP (j-1,...,p). Then the permutation matrix Pp,r is defined as

jth unit

nT n T1Pp,r -~~ Ieie j~ eje i 1-

J

Some elementary properties are

TPP~rPr.P- IrP PP~r- Pr~P

PP,I- Ip Pl,r- Ir

but their importance is due to the fact that for arbitrary matrices A(pxq)

and B (rxs) we have

A~B - Pr p(B~A)PQ, .

4. Rules of calculation

In the following A and B are arbitrary constant matrices and ~p and tv are

4

arbitrary differentiable matrix valued functions of ueC". When necessary theirdimensions are given explicitly, otherwise they follow from the context. Wehave

1) De(A'GB) - (In~A)ÍDe~P)B

2) D~(~P~B) - (D~t~)~B

. [x!3) When AeC and ~peC then Da(A~p) -[Do(Ip~p)](A~I,)

4 ) Product rule

D9(Y'w) - (Do~P)~ f (In~G)De~

5) Chain rule for scalar ~p and ~eCm

D.(~GoV~) - (D~V~T )( D.~G)

Wwhere Da denotes differentiation w.r.t. 1V.

So far the proofs are straight forward by looking at the components. As anexample of their use consider an arbitrary nxn matrix A. Then we have

D~(uTAu) 4(D~uTA)u f( Io~uTA)D~u 1(D~uT)Au f( In~uTA)vecIo -

- Au f vec(uTA) - Au f ATU .

In order to be able to differentiate all kinds of functions we need one morerule

6) D~ÍIp~UT) - PP,o

The proof follows immediately from 1) and the given properties of permutationmatríces i.e.

De(Ip~uT) - De[Pt,p(uT~IP)PP,n] - [D.(UT~Ip)]PP,n - [In~IP]PP,n- PP,n O

We are now in a position that we can calculate the quadruple matrix from the

5

characteristic function. As an illustration we calculate in the next section

the quadruple matrix for the multivariate normal distribution.

5. The Normal case

Suppose the random vector x has the n-variate normal distribution with

dispersion matrix E and expectation zero i.e. x~No(O,E). (It is allowed for E

to be singular). The characteristic function ~p is given by

T

~(u) - é~u Eu u~n

Applying the operator DóDeDéDa and taking u-0 yields after some calculationsthe following nice expression for the Quadruple matrix

QN - (vecE)(vecE)T f E~E f( In~E)Po n( In~E).

8. Application to Wishart matrices

I.et Y be a nxn random matrix with the Wishart distribution W,~ n(E) with m

degrees of freedom and dispersion matrix E. Then expectations of quadraticforms in Y such as ~Y(V)-E(YVYT) can simply be expressed in the quadruple

matrix QN of the underlying multinormal distribution. We have the followingresult

vec[~(V)J - (mQN f m(m-1)E~E]vec(V).

The proof is simple and straight forward and left to the reader.

6

References

Magnus, J.R. and Neudecker, H, (1988) Matrix Differential Calculus withApplications in Statistics and Econometrics, Wiley, New York.

7

i

IN 1992 REEDS vERSCHENEN

532 F.G. van den Heuvel en M.R.M, TurlingsPrivatisering van arbeidsongeschiktheidsregelingenRefereed by Prof.Dr. H. Verbon

533 J.C. Engwerda, L.G. van WilligenburgLQ-control of sampled continuous-time systemsRefereed by Prof.dr. J.M. Schumacher

534 J.C. Engwerda, A.C.M. Ran 8~ A.L. RijkeboerNecessary and sufficient conditions for the existence of a positivedefinite solution of the matrix equation X 4 A~`X-lA - Q.Refereed by Prof.dr. J.M. Schumacher

535 Jacob C. EngwerdaThe indefinite LQ-problem: the finite planning horizon caseRefereed by Prof.dr. J.M. Schumacher

536 Gert-Jan Otten, Peter Borm, Ton Storcken, Stef TijsEffectivity functions and associated claim game correspondencesRefereed by Prof.dr. P.H.M. Ruys

537 Jack P.C. Kleijnen, Gustav A. AlinkValidation of simulation models: mine-hunting case-studyRefereed by Prof.dr.ir. C.A.T. Takkenberg

538 V. Feltkamp and A, van den NouwelandControlled Communication NetworksRefereed by Prof.dr. S.H. Tijs

539 A, van SchaikProductivity, Labour Force Participation and the Solow Growth ModelRefereed by Prof.dr. Th.C.M.J. van de Klundert

540 J.J.G. Lemmen and S.C.W. EijffingerThe Degree of Financial Integration in the European CommunityRefereed by Prof.dr. A.B.T.M. van 5chaik

541 J. Bell, P.K. JagersmaInternationale Joint VenturesRefereed by Prof.dr. H.G. Barkema

542 Jack P.C. KleijnenVerification and validation of simulation modelsRefereed by Prof.dr.ir. C.A.T. Takkenberg

543 Gert NieuwenhuisUniform Approximations of the Stationary and Palm Distributionsof Marked Point ProcessesRefereed by Prof.dr. B.B. van der Genugten

11

544 R. Heuts, P. Nederstigt, W. Roebroek, W. SelenMulti-Product Cycling with Packaging in the Process IndustryRefereed by Prof.dr. F.A. van der Duyn Schouten545 J.C. Engwerda

Calculation of an approximate solution of the infinite time-varyingLQ-problemRefereed by Prof.dr. J.M. Schumacher

546 Raymond H.J.M. Gradus and Peter M. KortOn time-inconsistency and pollution control: a macroeconomic approachRefereed by Prof.dr. A.J. de Zeeuw

547 Drs. Dolph Cantrijn en Dr. Rezaul KabirDe Invloed van de Invoering van Preferente Beschermingsaandelen opAandelenkoersen van Nederlandse Beursgenoteerde OndernemingenRefereed by Prof.dr. P.W. Moerland

548 Sylvester Eijffinger and Eric SchalingCentral bank independence: criteris and indicesRefereed by Prof.dr. J.J. Sijben

549 Drs. A. SchmeitsGeYntegreerde investerings- en financieringsbeslissingen; Implicatiesvoor Capital BudgetingRefereed by Prof.dr. P.W. Moerland

550 Peter M. KortStandards versus standards: the effects of different pollutionrestrictions on the firm's dynamic investment policyRefereed by Prof.dr. F.A. van der Duyn Schouten

551 Niels G. Noorderhaven, Bart Nooteboom and Johannes BergerTemporal, cognitive and behavioral dimensions of transaction costs;to an understanding of hybrid vertical inter-firm relationsRefereed by Prof.dr. S.W. Douma

552 Ton Storcken and Harrie de SwartTowards an axiomatization of orderingsRefereed by Prof.dr. P.H.M. Ruys

553 J.H.J. RoemenThe derivation of a long term milk supply model from an optimizationmodelRefereed by Prof.dr. F.A. van der Duyn Schouten

554 Geert J. Almekinders and Sylvester C.W. EijffingerDaily Bundesbank and Federal Reserve Intervention and the ConditionalVariance Tale in DM~g-ReturnsRefereed by Prof.dr. A.B.T.M. van Schaik

555 Dr. M. Hetebrij, Drs. B.F.L. Jonker, Prof.dr. W.H.J. de Freytas"Tussen achterstand en voorsprong" de scholings- en personeelsvoor-zieningsproblematiek van bedrijven in de procesindustrieRefereed by Prof.dr. Th.M.M. Verhallen

lli

556 Ton GeertsRegularity and singularity in linear-quadratic control subject toimplicit continuous-time systemsCommunicated by Prof.dr. J. Schumacher

557 Ton GeertsInvariant subspaces and invertibility properties for singular sys-tems: the general caseCommunicated by Prof.dr. J. Schumacher

558 Ton GeertsSolvability conditions, consistency and weak consistency for lineardifferential-algebraic equations and time-invariant singular systems:the general caseCommunicated by Prof.dr. J. Schumacher

559 C. Fricker and M.R. JaibiMonotonicity and stability of periodic polling modelsCommunicated by Prof.dr.ir. O.J. Boxma

560 Ton GeertsFree end-point linear-quadratic control subject to implicit conti-nuous-time systems: necessary and sufficient conditions for solvabil-ityCommunicated by Prof.dr. J. Schumacher

561 Paul G.H. Mulder and Anton L. HempeniusExpected Utility of Life Time in the Presence of a Chronic Noncom-municable Disease StateCommunicated by Prof.dr. B.B. van der Genugten

562 Jan van der LeeuwThe covariance matrix of ARMA-errors in closed formCommunicated by Dr. H.H. Tigelaar

563 J.P.C. Blanc and R.D. van der MeíOptimization of polling systems with Bernoulli schedulesCommunicated by Prof.dr.ir. O.J. Boxma

564 B.B. van der GenugtenDensity of the least squares estimator in the multivariate linearmodel with arbitrarily normal variablesCommunicated by Prof.dr. M.H.C. Paardekooper

565 René van den Brink, Robert P. GillesMeasuring Domination in Directed GraphsCommunicated by Prof.dr. P.H.M. Ruys

566 Harry G. BarkemaThe significance of work incentives from bonuses: some new evidenceCommunicated by Dr. Th.E. Nijman

iv

567 Rob de Groof and Martin van TuijlCommercial integration and fiscal policy in interdependent, fínan-cially integrated two-sector economies with real and nominal wagerigidity.Communicated by Prof.dr. A.L. Bovenberg

568 F.A. van der Duyn Schouten, M.J.G. van Eijs, R.M.J. HeutsThe value of information in a fixed order quantity inventory systemCommunicated by Prof.dr. A.J.J. Talman569 E.N. Kertzman

Begrotingsnormering en EMUCommunicated by Prof.dr. J.W. van der Dussen

570 A. van den Elzen, D. TalmanFinding a Nash-equilibrium in noncooperative N-person games bysolving a sequence of linear stationary point problemsCommunicated by Prof.dr. S.H. Tijs

571 Jack P.C. KleijnenVerification and validation of modelsCommunicated by Prof.dr. F.A. van der Duyn Schouten

572 Jack P.C. Kleijnen and Willem van GroenendaalTwo-stage versus sequential sample-size determination in regressionanalysis of simulation experiments

573 Pieter K. JagersmaHet management van multinationale ondernemingen: de concernstructuur

574 A.L. HempeniusExplaining Changes in External Funds. Part One: TheoryCommunicated by Prof.Dr.Ir. A. Kapteyn

575 J.P.C. Blanc, R.D. van der MeiOptimization of Polling Systems by Means of Gradient Methodsand the Power-Series AlgorithmCommunicated by Prof.dr.ir. O.J. Boxma

576 Herbert HamersA silent duel over a cakeCommunicated by Prof.dr. S.H. Tijs

577 Gerard van der Laan, Dolf Talman, Hans KremersOn the existence and computation of an equilibrium in an economy withconstant returns to scale productionCommunicated by Prof.dr. P.H.M. Ruys

578 R.Th.A. Wagemakers, J.J.A. Moors, M.J.B.T. JanssensCharacterizing distributions by quantile measuresCommunicated by Dr. R.M.J. Heuts

v

579 J. Ashayeri, W.H.L. van Esch, R.M.J. HeutsAmendment of Heuts-Selen's Lotsizing and Sequencing Heuristic forSingle Stage Process Manufacturing SystemsCommunicated by Prof.dr. F.A. van der Duyn Schouten

580 H.G. BarkemaThe Impact of Top Management Compensation Structure on StrategyCommunicated by Prof.dr. S.W. Douma

581 Jos Benders en Freek AertsenAan de lijn of aan het lijntje: wordt slank produceren de mode?Communicated by Prof.dr. S.W. Douma

582 Willem HaemersDistance Regularity and the Spectrum of GraphsCommunicated by Prof.dr. M.H.C. Paardekooper

583 Jalal Ashayeri, Behnam Pourbabai, Luk van WassenhoveStrategic Marketing, Production, and Distribution Planning of anIntegrated Manufacturing SystemCommunicated by Prof.dr. F.A. van der Duyn Schouten

584 J. Ashayeri, F.H.P. DriessenIntegration of Demand Management and Production Planning in aBatch Process Manufacturing System: Case StudyCommunicated by Prof.dr. F.A. van der Duyn Schouten

585 J. Ashayeri, A.G.M. van Eijs, P. NederstigtBlending Modelling in a Process Manufacturing SystemCommunicated by Prof.dr. F.A. van der Duyn Schouten

586 J. Ashayeri, A.J. Westerhof, P.H.E.L. van AlstApplication of Mixed Integer Programming toA Large Scale Logistics ProblemCommunicated by Prof.dr. F.A. van der Duyn Schouten

587 P. Jean-Jacques HeringsOn the Structure of Constrained EquilibriaCommunicated by Prof.dr. A.J.J. Talman

V1

IN 1993 ~DS vEFtsc[~v~v

588 Rob de Groof and Martin van TuijlThe Twin-Debt Problem in an Interdependent WorldCommunicated by Prof.dr. Th. van de Klundert

Bibliotheek K. U. Brabantii f ~i~ 7 000 O1 ~ 73863 ~