Moments of Matrix Variate Skew Elliptically Contoured Distributions

Size: px

Start display at page:

Download "Moments of Matrix Variate Skew Elliptically Contoured Distributions"

Hilary Dennis
5 years ago
Views:

East Tennessee State Univesity Digital Coons @ East Tennessee State Univesity ETSU Faculty Woks Faculty Woks 3 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions Shiin Zheng East Tennessee

1 East Tennessee State Univesity Digital East Tennessee State Univesity ETSU Faculty Woks Faculty Woks 3 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions Shiin Zheng East Tennessee State Univesity; Nanjing Audit Univesity, zhengs@etsuedu Jeff Knisley East Tennessee State Univesity, knisleyj@etsuedu Chuning Zhang Univesity of Wisconsin Follow this and additional woks at: htts://dcetsuedu/etsu-woks Pat of the Statistics and Pobability Coons Citation Infoation Zheng, Shiin; Knisley, Jeff; and Zhang, Chuning 3 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions Advances and Alications in Statistics Vol36( 3-7 htt://wwwhjco/abstact/793ht ISSN: This Aticle is bought to you fo fee and oen access by the Faculty Woks at Digital East Tennessee State Univesity It has been acceted fo inclusion in ETSU Faculty Woks by an authoized adinistato of Digital East Tennessee State Univesity Fo oe infoation, lease contact digilib@etsuedu

2 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions Coyight Stateent Pusha Publishing House This docuent was ublished with eission by Pusha Publishing House It was oiginally ublished in Advances and Alications in Statistics This aticle is available at Digital East Tennessee State Univesity: htts://dcetsuedu/etsu-woks/77

Advances and Alications in Statistics 3 Pusha Publishing House, Allahabad, India Available online at htt://hjco/jounals/adasht Volue 36, Nube, 3, Pages 3-7 MOMENTS OF MATRIX VARIATE SKEW ELLIPTICALLY

3 Advances and Alications in Statistics 3 Pusha Publishing House, Allahabad, India Available online at htt://hjco/jounals/adasht Volue 36, Nube, 3, Pages 3-7 MOMENTS OF MATRIX VARIATE SKEW ELLIPTICALLY CONTOURED DISTRIBUTIONS Shiin Zheng,, Jeff Knisley 3 and Chuning Zhang 4 Institute fo Quantitative Biology Deatent of Biostatistics and Eideiology East Tennessee State Univesity Box 759, Johnson City, TN 3764, U S A Deatent of Finance Nanjing Audit Univesity Nanjing, P R China 3 Institute fo Quantitative Biology Deatent of Matheatics and Statistics East Tennessee State Univesity Box 7663, Johnson City, TN 3764, U S A 4 Deatent of Statistics Univesity of Wisconsin 3 Univesity Avenue Madison, WI 5376, U S A Abstact Matix vaiate skew ellitically contoued distibutions genealize seveal classes of iotant distibutions This ae defines and Received: May 3, 3; Acceted: June 4, 3 Matheatics Subject Classification: Piay 6H; Seconday 6H5 Keywods and hases: atix vaiate, skew ellitically contoued distibution, skew Peason tye VII distibution, skew noal distibution, stochastic eesentation, oent

4 4 Shiin Zheng, Jeff Knisley and Chuning Zhang exloes atix vaiate skew ellitically contoued distibutions In aticula, we discuss the fist two oents of the atix vaiate skew ellitically contoued distibutions Intoduction The oents of atix vaiate distibutions ae necessay both in theoy and in alications, but often, thei deivation is not staightfowad and can constitute a wok in and of itself Fo exale, Wen and Zhu deived stochastic eesentations and the fist two oents, aong othe oeties, of the ultivaiate skew Peason tye VII distibution and the skew t-distibution [, ] In addition, Chen and Guta extended seveal iotant oeties of the ultivaiate skew noal distibutions to the atix vaiate case [3] Also, Guta et al deived the fist two oents of the ultivaiate skew noal distibutions [4, 5, 6] Futheoe, Akdei and Guta discussed oe geneal atix vaiate skew noal distibutions, including oents [7] In a evious wok [8], we obtained the stochastic eesentations of the atix vaiate skew ellitically contoued distibutions (MSE This ae extends that wok by deiving the fist two oents of the atix vaiate skew ellitically contoued distibutions (MSE In Section, we ecall seveal definitions etinent to the ae as well as the esults fo [8] fo stochastic eesentations of atix vaiate skew ellitically contoued distibutions (MSE In Section 3, we obtain oents of atix vaiate skew ellitically contoued distibutions Notation and Definitions Matix vaiate distibutions have been studied by Guta and Naga, aong othes [9] Below we include the esults fo [8] necessay fo the deivation of oents We efe the eade to [9] and [8] fo additional definitions and theoes

5 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions 5 Let X be a ando atix, which has a atix vaiate noal distibution, ie, X ~ N, ( M, Σ Ψ, whee M R is a ean atix, Σ is a ositive definite atix and Ψ is an ositive definite atix We use bold face ue and lowe case lettes to denote vectos, uecase lettes without bold face fo atices, and lowecase lettes without bold face fo eleents of a vecto o a atix We list below the theoes and coollaies fo [8] which ae used in the following section Theoe (Theoe in [8] Assue V V ~ E, (, Σ Ψ, ψ, V Σ Σ Σ is, Ψ is, V is, V is Fo δ j (,, let X V, X j δ j V ( δ j Vj, V ( v, v,, v, Z ( X (, X, X X,, X, V V, V c Ψ ( Ψ h tx X dx X < j,,,, Then the ando atix X has a atix vaiate skew ellitically contoued distibution Secifically, X ~ MSE, ( Σδ δ, Σδ, Ψ, Ω Ψ, ψ,

6 6 Shiin Zheng, Jeff Knisley and Chuning Zhang whee δ ( δ, δ,, δ, Σδ δδ ΔΣΔ, Δ diag ( δ,, ( δ, and Ω δσ δ δ In the following coollaies, we use the notation and definitions found in Theoe V Coollay Assue V ~ N, (, Σ Ψ Then the ando V atix X has a atix vaiate skew noal distibution The df of the ando atix X can be witten as f X det( Ψ ( π ex Ψ < X X dx X det δδ ΔΣΔ det Ψ π ex t( X Σ Ψ δ X X < det( Ω det( Ψ ( π ex t(( X X ( X X δσ Ω δσ Ψ dx δ δ φ ( Σ Ψ φ X,, Ψ dx, X,, X < δ Φ δ ( X Σ δ,, Ω Ψ which is denoted by MSN, ( Σδ δ, Σδ, Ψ, Ω Ψ V Coollay Assue ~ PVII, (, Σ, Ψ,, Then the V ando atix X has a atix vaiate skew Peason tye VII distibution The df of the ando atix X can be witten as

7 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions 7 X Ψ X f X < X π Ψ dx ( ( π ( Σδ Ψ t( X Σδ X Ψ ( ( π ( X < ΩΨ [ X ] [ ] δ Σδ X ΩΨ X δ Σδ X dx This atix vaiate skew Peason tye VII distibution is denoted by MSPVII, ( Σδ δ, Σδ, Ψ, Ω Ψ,, In the next theoe, we use the sae notation and definitions as in Theoe Theoe (Theoe 3 in [8] Assue V V ~ E, (, Σ Ψ, ψ V Also assue that Σ A A, and Ψ B B Fo δ j (,, let X δ V V Δ, o X δ V ΔV X δ V Then the ando atix X has the following stochastic eesentation: X R( δ U ΔAU, ( B whee R is a nonnegative ando vaiable, U is and unifoly distibuted on shee S, U is and vec ( U is unifoly distibuted

8 8 Shiin Zheng, Jeff Knisley and Chuning Zhang on shee S In addition, R, U, and U ae indeendent Futheoe, ψ( x Ω ( x df(, x, whee Ω ( t t, t R denotes the chaacteistic function of vec ( U, and F ( denotes the distibution function of The next coollay follows iediately Moeove, its stateent uses the sae notation as in Theoe and also assues that Σ A A, Ψ BB let V Coollay 3 Assue V ~ N, (, Σ Ψ V Fo (,, δ j Z δ V V Δ, o Z δ V ΔV Z δ V Then the ando atix Z is distibuted as atix vaiate skew noal and has the following stochastic eesentation: Z R ( δ U ΔAU, ( B whee U, U, A, B ae the sae as in Theoe Based on the definitions and the stochastic eesentation obtained in [8], the oents of atix vaiate skew ellitically contoued distibution can be deived, as we now show 3 Moents of Matix Vaiate Skew Ellitically Contoued Distibution The oents of atix vaiate skew ellitical distibutions can be found using the oents of the atix vaiate skew noal distibution since the latte is easy to find Suose that the ando atix X is distibuted as a geneal atix vaiate skew ellitically contoued distibution (MCD such as in Theoe and also suose that the ando atix Z is distibuted as a atix vaiate skew noal as in Coollay 3 Then the oents of the skew

9 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions 9 MCD X ae associated with the oents of skew noal Z Secifically, we have the following elationshi between the: i j sij E R sij sij E, X i j ij E Z i j ij (3 i j sij E R whee X ij is the ( i, j eleent of ando atix X, Z ij is the ( i, j coefficient of the ando atix Z, and R is distibuted as χ with degees of feedo Euation (3 leads to the following esults: E( R E( R E ( X E( Z, E( XX E( ZZ, (4 E( R E( R ( E( R E vec X vec X E( vec( Z vec( Z (5 E( R It also leads to ou next theoe Theoe 3 Suose that the ando atix X is distibuted as a atix vaiate skew Peason tye VII defined in Coollay, ie, X ~ MSPVII, ( Σδ δ, δδ ΔΣΔ, Ψ, Ω Ψ,, Suose also that the ando atix Z is distibuted as a atix vaiate skew noal defined in Coollay, ie, Z MSN ( Σδ δ, Σδ, Ψ, Ω Then we have ~, Ψ ( E ( X E( Z, (6 ( E ( X X E( ZZ, (7 ( E [ vec( X vec( X ] E[ vec( Z vec( Z ] (8 (

10 Shiin Zheng, Jeff Knisley and Chuning Zhang Poof Based on (57 on 59 of [], the df of the ando vaiable R can be found using the function h defined in (6 of [8]: h g π π π Thus, d g R E d Next, we find the exectation of R By (57 on 59 of [], we have h g π

11 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions e π π Let s Then s e s s g Theefoe, S o R ~ χ and S R E E ds e s s, R R E E (9 On the othe hand,, g R E and, R E R R E E ( Hence, euations (6, (7 and (8 ae oved using (9 and (

12 Shiin Zheng, Jeff Knisley and Chuning Zhang In ode to use Theoe 3 to find the fist two oents of any atix vaiate skew ellitically contoued distibution, we have to fist find the fist two oents of atix vaiate skew noal distibutions Guta et al deived the fist two oents of ultivaiate skew noal distibutions [4, 5, 6] To genealize these esults to atix vaiate skew noal distibution, we utilize oent geneating functions In aticula, Aellano-Valle and Azzalini deived the oent geneating function (gf of a ultivaiate unified skew noal distibution [] Haa and Guta esented the gf fo geneal atix vaiate skew noal distibutions [] The following lea is fo Haa and Guta [] Lea Assue X ~ MSN, ( b, Σ, Ψ, Ω Then the oent geneating function of X can be witten as M X T c et ΣT Φ ( Ψ Σ, Ω Σ Ψ, T b b b ( whee c [ Φ ( ; Ω bσbψ ] In ode to silify notation in the seuel, we let Ξ Ω b ΣbΨ and θ Σb Note that θ vec ( Σb [( bσ Ψ] vec( T Futheoe, we let t vec ( T, D ( bσ Ψ Theoe 4 Suose X ~ MSN, ( b, Σ, Ψ, Ω Then the exectation of X can be witten as E ( X [ Φ ( ; Ξ ] G ( ; D, Ξ H, ( i j whee the atix H ij ( has unit eleent at the ( i, jth lace and zeo elsewhee and that Gij ( ; D, Ξ Φ ( ; Ξ Dt t T ij ij ij π Ξ D k, k k k ( i j ( Ξ Φ [ ; ( Ξ ],

13 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions 3 whee D k ( i j, is the eleent ( k, ( i j of atix D, Ξ ( k is the atix constucted by eliinating the kth ow and the kth colun of Ξ, with the convention that Ξ ( Ξ ( k k t( ΣT Poof Since ΣT Ψ, the deivative with esect to T of T the oent geneating function ( can be witten in the fo M X ( T c et ΣT ( Σ Ψ Φ ( Ξ T θ, T c et ΣT Φ (, Ξ θ T (3 Afte a lengthy calculation (the eade is efeed to siila deivations in [4, 5, 6] fo details, we have E ( X M XT ( T T [ Φ( ; Ξ ] Φ (, Ξ, T θ T whee T Φ ( θ, Ξ Φ ( ; Ξ Dt T T T and T Φ ( Dt ; Ξ G ( ; D, Ξ H T i j ij ij Theefoe, the theoe is oved Finally, we have the following theoe, whose oof like that of Theoe 4 euies a long eleentay calculation of a second deivative which is siila in details to deivations in [4, 5, 6]

14 4 Shiin Zheng, Jeff Knisley and Chuning Zhang Theoe 5 Suose X ~ MSN, ( b, Σ, Ψ, Ω Then the exectation of vec ( X vec( X can be witten as E ( vec( X vec( X Σ Ψ [ Φ ( ; Ξ ] G ( ; D, Ξ H i j whee the atix ( has unit eleent at the ( i, j th lace and H i j i j zeo elsewhee, i ( i j, j ( a b; i, a,, ; j, b,, ; and (, D Ξ G i j, i j, Φ ( Dt, Ξ t t ij ab t π D l D, Ξ k l k k, l k, l ( i j k, ( a b Ξ Φ( ; [ Ξ ] whee k i i k ki Dl, ( i jdk, ( a bξ xi, φ k k k k k kk [ x, x, x,, x ; ( Ξ ] dx dx dx dx, Ξ is the eleent ( k, k of Ξ and Ξ ( i, j is the atix constucted by eliinating the ith and jth ows and the ith and jth coluns of Ξ, with Ξ ( Ξ (, i, j i j Poof Based on (, we have

15 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions 5 Σ Ψ Φ t T T M X T Σ Ψ c et T T vec T vec( T c et ΣT Φ (, Ξ θ vec( T Thus, M X ( T vec( T vec( T Σ Ψ Σ Ψ t T T t T T Φ c et ΣT vec( T vec( T Σ Ψ t T T Φ ( Ξ c et ΣT θ, ( ( vec T vec T Σ Ψ Φ Ξ ( t T T c et ΣT θ, vec T vec T Φ ( Ξ c et ΣT θ, vec( T vec( T ( θ, Ξ ( c et ΣT Ψ Σ Ψ Σ Φ ( Ξ vec T vec T θ, Φ ( Ξ ( Σ Ψ Φ ( Ξ θ, ( vec( Σ θ, vec T ( θ, Ξ Φ (, θ Ξ vec( T vec( T

16 6 Shiin Zheng, Jeff Knisley and Chuning Zhang Hence, E [ vec( X vec( X ] M X ( T vec( T vec( T T Φ (, Ξ Σ Ψ θ c vec( T vec( T T 4 Concluding Reaks The set of skew ellitically contoued distibutions contains the ellitically contoued distibutions In [8], we defined and exloed the atix vaiate skew ellitically contoued distibution, obtaining in the ocess two stochastic eesentations In this wok, we have extended that effot by obtaining the fist two oents of a subfaily of this lage faily of distibutions In aticula, the oent esults obtained in this ae ae useful in both theoy and alication Futue diections include exanding uon the utility and iotance of these esults Acknowledgents The authos gaciously acknowledge the financial and collaboative suot of the Institute fo Quantitative Biology at East Tennessee State Univesity and Zhang s eseach is suoted by the US NSF gant DMS and Wisconsin Aluni Reseach Foundation The authos would also like to thank the efeees fo any useful suggestions and coents which ioved the esentation in this ae Refeences [] Y J Wen and D Y Zhu, New ultivaiate skew t distibutions geneated fo skew Peason vii distibutions, Chinese Jounal of Pobability and Statistics 5(5 (9,

17 Moents of Matix Vaiate Skew Ellitically Contoued Distibutions 7 [] Y J Wen and D Y Zhu, Poeties of new ultivaiate skew t distibutions geneated fo skew Peason vii distibutions, Chinese Jounal of Pobability and Statistics 6(6 (, [3] J T Chen and A K Guta, Matix vaiate skew noal distibutions, Statistics 39(3 (5, [4] G González-Faias, J A Do inguez- Molina and A K Guta, A geneal ultivaiate skew noal distibution, Technical Reot -9, Deatent of Matheatics and Statistics, Bowling Geen State Univesity, [5] A K Guta, G González-Faias and J A Do inguez- Molina, A ultivaiate skew noal distibution, Technical Reot No -, Deatent of Matheatics and Statistics, Bowling Geen State Univesity, [6] A K Guta, G González-Faias and J A Do inguez- Molina, A ultivaiate skew noal distibution, Jounal of Multivaiate Analysis 89 (4, 8-9 [7] D Akdei and A K Guta, A atix vaiate skew distibution, Euoean Jounal of Pue and Alied Matheatics 3( (, 8-4 [8] S Zheng, C Zhang and J Knisley, Stochastic eesentations of the atix vaiate skew ellitically contoued distibutions, Advances and Alications in Statistics 33( (3, [9] A K Guta and D K Naga, Matix Vaiate Distibutions, Chaan & Hall/CRC, Boca Raton, London, New Yok, Washington, DC, [] K T Fang and Y T Zhang, Genealized Multivaiate Analysis, Singe-Velag, Belin Heidelbeg, New Yok, London, Pais, Tokyo, Hong Kong, 99 [] R B Aellano-Valle and A Azzalini, On the unification of failies of skewnoal distibutions, Scand J Statist 33 (6, [] S Haa and A K Guta, On atix vaiate skew noal distibution, Statistics 4 (8, 79-94

Some Remarks on the Boundary Behaviors of the Hardy Spaces

Some Remarks on the Boundary Behaviors of the Hardy Spaces Soe Reaks on the Bounday Behavios of the Hady Spaces Tao Qian and Jinxun Wang In eoy of Jaie Kelle Abstact. Soe estiates and bounday popeties fo functions in the Hady spaces ae given. Matheatics Subject