Seite - 243 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 whichtogetherwith (30) tellsus that: ΔVs (x)= x s1 11Δ V′ s′ (x ′−x−111UtU), (31) where s′ :=(s2, . . . ,sr)∈Cr−1. Letusconsider theadjointmapτ∗B :ZV→ZV ofτBwithrespect to thestandardinnerproduct. Let b ∈ Rq1 be the vector corresponding to B ∈W. For x ∈ ZV and ξ ∈ ZV as in (3) and (10), respectively,weobserve that: (τBx|ξ)= x11ξ11+2t(u+x11b)v+(x′+UtB+BtU+x11BtB|ξ′) = x11(ξ11+2tbv+ tbψ(ξ′)b)+2tu(v+ψ(ξ′)b)+(x′|ξ′). Thus, ifwewrite: ι(ξ11,v,ξ′) := ( ξ11In1 tV V ξ′ ) , wehave: τ∗Bι(ξ11,v,ξ′)= ι(ξ11+2tbv+ tbψ(ξ′)b,v+ψ(ξ′)b,ξ′). (32) Furthermore,wesee from(12) thatφ1(τ∗Bι(ξ11,v,ξ′))equals:( ξ11+2tbv+ tbψ(ξ′)b tv+ tbψ(ξ′) v+ψ(ξ′)b ψ(ξ′) ) = ( 1 tb Iq1 )( ξ11 tv v ψ(ξ′) )( 1 b Iq1 ) , so thatweget forξ= ι(ξ11,v,ξ′): φ1(τ ∗ Bξ)= ( 1 tb Iq1 ) φ1(ξ) ( 1 b Iq1 ) . Therefore: (φ1(τ ∗ Bξ) −1)11=(φ1(ξ)−1)11. Ontheotherhand,wehaveforξ= ι(ξ11,v,ξ′)∈P∗V: δVs (ξ)=(φ1(ξ)−1) −s1 11 δ V′ s′ (ξ ′). (33) Thus,weconcludethat: δVs (τ∗Bξ)= δVs (ξ). (34) Theorem3. When sk>−1−qk/2 for k=1,. . . ,r, onehas:∫ PV e−(x|ξ)ΔVs (x)dx=C−1V γV(s)δ V−s(ξ)ϕV(ξ), (35) whereγV(s) :=(2π)(N−r)/2∏rk=1Γ(sk+1+ qk 2 ). Proof. RecallingTheorem2,werewrite theright-handsideof (35)as: (2π)(N−r)/2 r ∏ k=1 Γ(sk+1+ qk 2 ) r ∏ k=1 ( φk(ξ) −1)sk+1+qk/2 11 ∏ qk>0 (detψk(ξ)) −1/2, 243