Page - 246 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 When s1>−2, s2>−3/2and s3>−1, the integral formula (35)holdswith: γV(s)=(2π)3/2Γ(s1+2)Γ(s2+3/2)Γ(s3+1). Furthermore,when s1>1, s2>1/2and s3>0, the integral formula (37)holdswith: ΓV(s)=(2π)3/2Γ(s1−1)Γ(s2−1/2)Γ(s3). 5.MultiplicativeLegendreTransformofGeneralizedPowerFunctions Fors∈Rr>0,weseethat logΔ−s isastrictlyconvexfunctionontheconePV. Infact,Δ−s isdefined naturallyonPn asaproductofpowersofprincipalminors,andit iswellknownthatsuchlogΔ−s is strictlyconvexonthewholePn. In thissection,weshall showthat logΔV−sandlogδV−sarerelatedby theFenchel–Legendre transform. Forx∈PV,wedenotebyIVs (x) theminusgradient−∇logΔ−s(x)atxwithrespect to the inner product.Namely,IVs (x) isanelementofZV forwhich: (IVs (x)|y)=− (d dt ) t=0 logΔ−s(x+ ty) (y∈ZV). Similarly,JVs (ξ) :=−∇logδ−s(ξ) isdefinedforξ∈P∗V. Ifq1>0, thenforanyB∈W,wehave: IVs ◦τB=τ∗B◦IVs , (41) JVs ◦τ∗B=τB◦JVs (42) owingto (30)and(34), respectively. Theorem5. For any s∈Rr>0, themapIVs :PV→ZV gives adiffeomorphism fromPV ontoP∗V, andJVs gives the inversemap. Proof. Weshallprove the statementby inductionon the rank. When r= 1,wehaveIVs (x11In1)= s1 x11 In1 =JVs (x11In1) forx11>0. Thus, thestatement is true in thiscase. When r>1,assumethat thestatementholds for thesystemofrank r−1. LetZ0V be thesubspace ofZV definedby: Z0V := {( x11In1 0 0 x′ ) ; x11∈R, x′ ∈ZV′ } . Bydirect computationwith (31)and(33),wehave: IVs ( x11In1 0 0 x′ ) = ( s1 x11 In1 0 0 IV′s′ (x′) ) , (43) JVs ( ξ11In1 0 0 ξ′ ) = ( s1 ξ11 In1 0 0 JV′s′ (ξ′) ) (44) for x11, ξ11> 0, x′ ∈ PV′ and ξ′ ∈ P∗V′. By the inductionhypothesis,we see thatIVs :PV∩Z0V → P∗V∩Z0V isbijectivewith the inversemapJVs :P∗V∩Z0V→PV∩Z0V. Ifq1=0, thestatementholdsbecauseZV=Z0V. Assumeq1>0. Lemma1(ii) tellsus that, for x∈PV, thereexistuniquex0∈Z0V∩PV andB∈W forwhichx=τBx0. Similarly,wesee from(32) that, for ξ ∈P∗V, there exist unique ξ0 ∈Z0V∩P∗V andC∈W forwhich ξ = τ∗Cξ0. Therefore,we deduce from(41)and(42) thatIVs :PV →P∗V isabijectionwithJVs the inversemap. 246