The Fisher information matrices are derived for three of the most popular elliptically symmetric distributions: the Pearson type II, Pearson type VII and the Kotz type distributions. We hope the results could be important to the many researchers working in this area

Kotz, Samuel

Nadarajah, Saralees

The Fisher information matrices are derived for three of the most popular elliptically symmetric distributions: the Pearson type II, Pearson type VII and the Kotz type distributions. We hope the results

could be important to the many researchers working in this area.Peer Reviewe

UPCommons

Information matrices for some elliptically symmetric distributions

The Fisher information matrices are derived for three of the most popular elliptically symmetric distributions: the Pearson type II, Pearson type VII and the Kotz type distributions. We hope the results
could be important to the many researchers working in this area.Peer Reviewe

UPCommons. Portal del coneixement obert de la UPC

Statistics & Operations Research TransactionsSORT 29 (1) January-June 2005, 43-56Statistics &Operations ResearchTransactionsInformation matrices for some elliptically symmetricdistributionsc© Institut d’Estadı´stica de Catalunyasort@idescat.esISSN: 1696-2281www.idescat.net/sortSaralees Nadarajah1, Samuel Kotz21 University of Nebraska, 2 The George Washington UniversityAbstractThe Fisher information matrices are derived for three of the most popular elliptically symmetricdistributions: the Pearson type II, Pearson type VII and the Kotz type distributions. We hope the resultscould be important to the many researchers working in this area.MSC: 33C90, 62E99Keywords: Elliptically symmetric Kotz type distribution, Elliptically symmetric Pearson type IIdistribution, Elliptically symmetric Pearson type VII distribution, Fisher information matrices1 IntroductionThe elliptically symmetric Pearson type II, Pearson type VII and the Kotz typedistributions are given by the joint pdfsf (x, y) = N + 1π√1 − ρ2(1 − x2 + y2 − 2ρxy1 − ρ2)N(1)(for N > −1 and −1 < ρ < 1),∗Address for correspondence: Saralees Nadarajah, Department of Statistics, University of Nebraska, Lincoln, NE68583.Samuel Kotz, Department of Engineering Management and Systems Engineering, The George WashingtonUniversity, Washington, D.C. 20052.Received: September 2004Accepted: December 200444 Information matrices for some elliptically symmetric distributionsf (x, y) = N − 1πm√1 − ρ2⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−N(2)(for N > 1, m > 0 and −1 < ρ < 1), andf (x, y) =srN/s(x2 + y2 − 2ρxy)N−1πΓ (N/s)(1 − ρ2)N−1/2 exp{−r(x2 + y2 − 2ρxy1 − ρ2)s}(3)(for N > 0, r > 0, s > 0 and −1 < ρ < 1), respectively. The bivariate t-distribution andthe bivariate Cauchy distribution are special cases of (2) for N = (m + 2)/2 and m = 1,N = 3/2, respectively. When s = 1, (3) is the original Kotz distribution introduced inKotz (1975). When N = 1, s = 1 and r = 1/2, (3) reduces to a bivariate normal density.The parameter ρ is the correlation coeﬃcient between the x and y components. Fordetails on properties and applications of these distributions see Johnson (1987), Fang etal. (1990), Nadarajah (2003) and Kotz and Nadarajah (2004).The aim of this note is to calculate the Fisher information matrices corresponding toeach of the pdfs given by (1), (2) and (3). This requires calculation of product momentsof the form E(XmYn). A transformation which aides this task is:(XY)=12( √1 + ρ +√1 − ρ √1 + ρ − √1 − ρ√1 + ρ − √1 − ρ √1 + ρ + √1 − ρ) (UV). (4)Under this transformation, one can easily see that (1), (2) and (3) reduce tof (u, v) = N + 1π(1 − u2 − v2)N, (5)f (x, y) = N − 1πm(1 +u2 + v2m)−N, (6)andf (u, v) =srN/s(x2 + y2)N−1πΓ (N/s) exp{−r(x2 + y2)s}, (7)respectively. Furthermore, the product moments for (5), (6) and (3) are given byE (U pVq) = N + 1πB(N + 1,p + q2+ 1)B(p + 12,q + 12), (8)Saralees Nadarajah, Samuel Kotz 45E (U pVq) = m(p+q)/2(N − 1)πB( p + q2+ 1,N − p + q2− 1)B(p + 12,q + 12), (9)andE (U pVq) =Γ(2N + p + q2s)πr(p+q)/(2s)Γ (N/s) B(p + 12,q + 12), (10)respectively, if both p ≥ 0 and q ≥ 0 are even integers (see, for example, Fang et al.(1990)). If either p or q is odd then the moment equals to zero. Hence, the productmoment E(XmYn) for any m and n can be calculated by combining (4), (8), (9) and (10).Certain relationships which express E(XmYn) in terms of product moments of (U,V) aregiven in the Appendix (Section 5).The calculations for the Kotz type distribution require additional moments of theformJ(m, n) = E[(X2 + Y2 − 2ρXY1 − ρ2)m {log(X2 + Y2 − 2ρXY1 − ρ2)}n](11)andK(m, n) = E[XY(X2 + Y2 − 2ρXY1 − ρ2)m {log(X2 + Y2 − 2ρXY1 − ρ2)}n]. (12)For this, apply the transformation(X,Z) =(X,X2 + Y2 − 2ρXY1 − ρ2).Note that the jacobian of this transformation is:| J | =√1 − ρ22√Z − X2.Thus, (11) and (12) can be reduced toJ(m, n) = 2srN/sP(m, n)πΓ (N/s)(1 − ρ2)N−1 (13)andK(m, n) = 2srN/sρQ(m, n)πΓ (N/s)(1 − ρ2)N−1 , (14)46 Information matrices for some elliptically symmetric distributionsrespectively, where P(m, n) and Q(m, n) denote the integralsP(m, n) =∫ ∞0∫ √z0zN+m−1(log z)n exp (−rzs)√z − x2dxdzandQ(m, n) =∫ ∞0∫ √z0x2zN+m−1(log z)n exp (−rzs)√z − x2dxdz.Integrating with respect to the x variable yieldsP(m, n) = π2∫ ∞0zN+m−1(log z)n exp (−rzs) dzandQ(m, n) = π4∫ ∞0zN+m(log z)n exp (−rzs) dz.These integrals can be calculated by using equation (2.6.21.1) in Prudnikov et al. (1986,volume 1) to yieldP(m, n) = π2s(∂∂α)n [r−α/sΓ(αs)]∣∣∣∣∣∣α=N+m(15)andQ(m, n) = π4s(∂∂α)n [r−α/sΓ(αs)]∣∣∣∣∣∣α=N+m+1. (16)Hence, (11) and (12) can be calculated by combining (13)–(14) and (15)–(16).The exact forms of the information matrices are given in Sections 2, 3 and 4. Thecalculations use the digamma function defined by Ψ(x) = d logΓ(x)/dx.2 Information matrix for Pearson IIIf (x, y) is a single observation from (1) then the log-likelihood function can be writtenaslog L(N, ρ) = log(N + 1) − log π − 12log(1 − ρ2)+ N log(1 − x2 + y2 − 2ρxy1 − ρ2).Saralees Nadarajah, Samuel Kotz 47The first-order derivatives are:∂ log L∂N=1N + 1+ log(1 − x2 + y2 − 2ρxy1 − ρ2)and∂ log L∂ρ=ρ1 − ρ2 −2N{ρx2 + ρy2 −(1 + ρ2)xy}(1 − ρ2)2(1 − x2 + y2 − 2ρxy1 − ρ2)−1.The second-order derivatives are:∂2 log L∂N2= − 1(N + 1)2 ,∂2 log L∂N∂ρ= −2{ρx2 + ρy2 −(1 + ρ2)xy}(1 − ρ2)2(1 − x2 + y2 − 2ρxy1 − ρ2)−1,and∂2 log L∂ρ2=1 + ρ2(1 − ρ2)2 −4N{ρx2 + ρy2 −(1 + ρ2)xy}2(1 − ρ2)4(1 − x2 + y2 − 2ρxy1 − ρ2)−2−2N[(1 − ρ2) (x2 + y2 − 2ρxy)+ 4ρ{ρx2 + ρy2 −(1 + ρ2)xy}](1 − ρ2)3 .Now, we can compute the elements of the Fisher information matrix. It is clear thatE(−∂2 log L∂N2)=1(N + 1)2 .By applying (8) and (17)–(19), one getsE(−∂2 log L∂N∂ρ)=(N + 1)B(2,N)ρ1 − ρ2 .48 Information matrices for some elliptically symmetric distributionsFinally, application of (8) and (17)–(24) yieldsE(−∂2 log L∂ρ2)=12(1 − ρ2)2{2N(N + 1)B(3,N − 1)ρ2 + 4N(N + 1)B(2,N)ρ2 − 2ρ2+N(N + 1)B(3,N − 1) + 4N(N + 1)B(2,N) − 2}.3 Information matrix for Pearson VIIIf (x, y) is a single observation from (2) then the log-likelihood function can be writtenaslog L(N,m, ρ) =log(N − 1) − log π − log m − 12log(1 − ρ2)− N log⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠ .The first-order derivatives are:∂ log L∂N=1N − 1 − log⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠ ,∂ log L∂m=N(x2 + y2 − 2ρxy)m2(1 − ρ2)⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1− 1m,and∂ log L∂ρ=ρ1 − ρ2 −2N{ρx2 + ρy2 −(1 + ρ2)xy}m(1 − ρ2)2⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1.The second-order derivatives are:∂2 log L∂N2= − 1(N − 1)2 ,∂2 log L∂N∂m=x2 + y2 − 2ρxym2(1 − ρ2)⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1,Saralees Nadarajah, Samuel Kotz 49∂2 log L∂N∂ρ= −2{ρx2 + ρy2 −(1 + ρ2)xy}(1 − ρ2)2⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1,∂2 log L∂m2=1m2−2N(x2 + y2 − 2ρxy)m3(1 − ρ2)⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1+N(x2 + y2 − 2ρxy)2m4(1 − ρ2)2⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−2,∂2 log L∂m∂ρ=2N{ρx2 + ρy2 −(1 + ρ2)xy}m2(1 − ρ2)2⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1−2N(x2 + y2 − 2ρxy) {ρx2 + ρy2 −(1 + ρ2)xy}m3(1 − ρ2)3⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−2and∂2 log L∂ρ2=1 + ρ2(1 − ρ2)2 +2N[(1 − ρ2) (x2 + y2 − 2ρxy)+ 4ρ{ρx2 + ρy2 −(1 + ρ2)xy}]m2(1 − ρ2)3×⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1×⎡⎢⎢⎢⎢⎢⎢⎢⎣4{ρx2 + ρy2 −(1 + ρ2)xy}2 ⎛⎜⎜⎜⎜⎜⎜⎝1 + x2 + y2 − 2ρxym(1 − ρ2)⎞⎟⎟⎟⎟⎟⎟⎠−1− m⎤⎥⎥⎥⎥⎥⎥⎥⎦ .Now, we can compute the elements of the Fisher information matrix. It is clear thatE(−∂2 log L∂N2)=1(N − 1)2 .By applying (9) and (17)–(19), one getsE(−∂2 log L∂N∂m)= − (N − 1)B(2,N − 1)m50 Information matrices for some elliptically symmetric distributionsandE(−∂2 log L∂N∂ρ)=(N − 1)B(2,N − 1)mρ1 − ρ2 .By applying (9) and (17)–(24), one getsE(−∂2 log L∂m2)=2N(N − 1)B(2,N − 1) + N(1 − N)B(3,N − 1) − 1m2andE(−∂2 log L∂m∂ρ)=N(N − 1) {B(3,N − 1) − B(2,N − 1)} ρm(1 − ρ2) .Finally, application of (9), (17)–(19) and (25)–(31) yieldsE(−∂2 log L∂ρ2)=1(1 − ρ2)2{2N(1 − N)B(4,N − 2)mρ8 + 3N(1 − N)B(4,N − 2)mρ6+11N(N − 1)B(4,N − 2)mρ4 + 5N(1 − N)B(4,N − 2)mρ2+2N(N − 1)B(2,N − 1)ρ2 − ρ2 + 2N(N − 1)B(2,N − 1)+N(1 − N)B(4,N − 2)m − 1}.4 Information matrix for Kotz typeIf (x, y) is a single observation from (3) then the log-likelihood function can be writtenaslog L(N, r, s, ρ) = log s + N log rs− log π − logΓ(Ns)+(12− N)log(1 − ρ2)+(N − 1) log(x2 + y2 − 2ρxy)− r(x2 + y2 − 2ρxy1 − ρ2)s.The first-order derivatives are:∂ log L∂N=log rs− 1sΨ(Ns)− log(1 − ρ2)+ log(x2 + y2 − 2ρxy),Saralees Nadarajah, Samuel Kotz 51∂ log L∂r=Nrs−(x2 + y2 − 2ρxy1 − ρ2)s,∂ log L∂s=1s− N log rs2+Ns2Ψ(Ns)− r(x2 + y2 − 2ρxy1 − ρ2)slog(x2 + y2 − 2ρxy1 − ρ2)and∂ log L∂ρ=(2N − 1)ρ1 − ρ2 −2(N − 1)xyx2 + y2 − 2ρxy−2rs{ρx2 + ρy2 −(1 + ρ2)xy}(1 − ρ2)2(x2 + y2 − 2ρxy1 − ρ2)s−1.The second-order derivatives are:∂2 log L∂N2= − 1s2Ψ′(Ns),∂2 log L∂N∂r=1rs,∂2 log L∂N∂s= − log rs2+1s2Ψ(Ns)+Ns3Ψ′(Ns),∂2 log L∂N∂ρ=2ρ1 − ρ2 −2xyx2 + y2 − 2ρxy ,∂2 log L∂r2= − Nr2s,∂2 log L∂r∂s= − Nrs2−(x2 + y2 − 2ρxy1 − ρ2)slog(x2 + y2 − 2ρxy1 − ρ2),∂2 log L∂r∂ρ= −2s{ρx2 + ρy2 −(1 + ρ2)xy}(1 − ρ2)2(x2 + y2 − 2ρxy1 − ρ2)s−1,52 Information matrices for some elliptically symmetric distributions∂2 log L∂s2= − 1s2+2N log rs3− 2Ns3Ψ(Ns)− N2s4Ψ′(Ns)−r{log(x2 + y2 − 2ρxy1 − ρ2)}2 (x2 + y2 − 2ρxy1 − ρ2)s,∂2 log L∂s∂ρ= −2r{ρx2 + ρy2 −(1 + ρ2)xy}(1 − ρ2)2(x2 + y2 − 2ρxy1 − ρ2)s−1×{1 + s log(x2 + y2 − 2ρxy1 − ρ2)},and∂2 log L∂ρ2=(2N − 1)(1 + ρ2)(1 − ρ2)2 − 4(N − 1)x2y2(x2 + y2 − 2ρxy)2 −4rs(s − 1){ρx2 + ρy2 −(1 + ρ2)xy}2(1 − ρ2)4×(x2 + y2 − 2ρxy1 − ρ2)s−2−2rs[(1 − ρ2) (x2 + y2 − 2ρxy)+ 4ρ{ρx2 + ρy2 −(1 + ρ2)xy}](1 − ρ2)3×(x2 + y2 − 2ρxy1 − ρ2)s−1.Now, we can compute the elements of the Fisher information matrix. It is clear thatE(−∂2 log L∂N2)=1s2Ψ′(Ns).E(−∂2 log L∂N∂r)= − 1rs,E(−∂2 log L∂N∂s)=log rs2− 1s2Ψ(Ns)− Ns3Ψ′(Ns),andE(−∂2 log L∂r2)=Nr2s.Saralees Nadarajah, Samuel Kotz 53By applying (10) and (19), one getsE(−∂2 log L∂N∂ρ)= − ρ1 − ρ2 .By applying (10) and (17)–(19), one getsE(−∂2 log L∂r∂ρ)=Nρr(1 − ρ2) .By applying (10) and (17)–(24), one getsE(−∂2 log L∂ρ2)=1 + Ns(1 + 2ρ2)2(1 − ρ2)2 .By applying (13), one getsE(−∂2 log L∂r∂s)=Nrs2+N{Ψ(1 +Ns)− log r}rs2(1 − ρ2)N−1andE(−∂2 log L∂s2)=1s2− 2N log rs3+2Ns3Ψ(Ns)+N2s4Ψ′(Ns)+N{Ψ′(1 +Ns)+ Ψ2(1 +Ns)− 2 log rΨ(1 +Ns)+ (log r)2}s3(1 − ρ2)N−1 .Finally, application of (17)–(19), (13) and (14) yieldsE(−∂2 log L∂s∂ρ)=Nρs(1 − ρ2) + 2Nρ{Ψ(1 +Ns)− log r}s(1 − ρ2)N −Nρ{Ψ(1 +Ns)− log r}s(1 − ρ2)N+1 .5 AppendixThe following relationships are needed for the calculation of the elements of the Fisherinformation matrices. These relations follow directly from the transformation (4).54 Information matrices for some elliptically symmetric distributionsE(X2)=1 +√1 − ρ22E(U2)+1 −√1 − ρ22E(V2), (17)E(Y2)=1 −√1 − ρ22E(U2)+1 +√1 − ρ22E(V2), (18)E (XY) = ρ2{E(U2)+ E(V2)}, (19)E(X4)=2√1 − ρ2 + 2 − ρ24E(U4)−2√1 − ρ2 − 2 + ρ24E(V4)+3ρ22E(U2V2), (20)E(Y4)=2√1 − ρ2 + 2 − ρ24E(V4)−2√1 − ρ2 − 2 + ρ24E(U4)+3ρ22E(U2V2), (21)E(XY3)=ρ{1 −√1 − ρ2}4E(U4)+ρ{1 +√1 − ρ2}4E(V4)+3ρ2E(U2V2), (22)E(X3Y)=ρ{1 −√1 − ρ2}4E(V4)+ρ{1 +√1 − ρ2}4E(U4)+3ρ2E(U2V2), (23)E(X2Y2)=ρ24{E(U4)+ E(V4)}+(1 +ρ22)E(U2V2), (24)E(X6)=4 − 3ρ2 +(4 − ρ2) √1 − ρ28 E(U6)+4 − 3ρ2 −(4 − ρ2) √1 − ρ28 E(V6)+15ρ2{1 −√1 − ρ2}8 E(U2V4)+15ρ2{1 +√1 − ρ2}8 E(U4V2), (25)Saralees Nadarajah, Samuel Kotz 55E(Y6)=4 − 3ρ2 +(4 − ρ2) √1 − ρ28 E(V6)+4 − 3ρ2 −(4 − ρ2) √1 − ρ28 E(U6)+15ρ2{1 −√1 − ρ2}8 E(U4V2)+15ρ2{1 +√1 − ρ2}8 E(U2V4), (26)E(XY5)=ρ{2 − ρ2 − 2√1 − ρ2}8 E(U6)+ρ{2 − ρ2 + 2√1 − ρ2}8 E(V6)+5ρ{2 + ρ2 + 2√1 − ρ2}8 E(U2V4)+5ρ{2 + ρ2 − 2√1 − ρ2}8 E(U4V2), (27)E(X5Y)=ρ{2 − ρ2 − 2√1 − ρ2}8 E(V6)+ρ{2 − ρ2 + 2√1 − ρ2}8 E(U6)+5ρ{2 + ρ2 + 2√1 − ρ2}8 E(U4V2)+5ρ{2 + ρ2 − 2√1 − ρ2}8 E(U2V4), (28)E(X2Y4)=ρ2{1 −√1 − ρ2}8 E(U6)+ρ2{1 +√1 − ρ2}8 E(V6)+4 + 11ρ2 +(4 + ρ2) √1 − ρ28 E(U2V4)+4 + 11ρ2 −(4 + ρ2) √1 − ρ28 E(U4V2), (29)56 Information matrices for some elliptically symmetric distributionsE(X4Y2)=ρ2{1 −√1 − ρ2}8 E(V6)+ρ2{1 +√1 − ρ2}8 E(U6)+4 + 11ρ2 +(4 + ρ2) √1 − ρ28 E(U4V2)+4 + 11ρ2 −(4 + ρ2) √1 − ρ28 E(U2V4), (30)andE(X3Y3)=ρ38{E(U6)+ E(V6)}+3ρ(4 + ρ2)8{E(U2V4)+ E(U4V2)}. (31)7 ReferencesFang, K. -T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. London:Chapman and Hall.Johnson, M. E. (1987). Multivariate Statistical Simulation. New York: John Wiley and Sons.Kotz, S. (1975). Multivariate distributions at a cross-road. In Statistical Distributions in Scientific Work, 1,editors G. P. Patil, S. Kotz and J. K. Ord. Dordrecht (eds)., The Netherlands: D. Reidel PublishingCompany.Kotz, S. and Nadarajah, S. (2004). Multivariate t Distributions and Their Applications. New York:Cambridge University Press.Nadarajah, S. (2003). The Kotz type distribution with applications. Statistics, 37, 341–358.Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series (volumes 1, 2 and 3).Amsterdam: Gordon and Breach Science Publishers.

Information matrices for some elliptically symmetric distributions

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