Page - 245 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 Therefore,by inductionontherank, it suffices toshowthat the left-handsideof (38)equals: (2π)q1/2Γ(s1−q1/2)x−s111 ∫ P∗V′ e−(x˜ ′|ξ′)δV ′ s′ (ξ ′)ϕV′(ξ′)dξ′. (39) Whenq1=0, sincedξ= dξ11dξ′, the left-handsideof (38)equals:∫ ∞ 0 e−x11ξ11ξs1−111 dξ11 ∫ P∗V′ e−(x ′|ξ′)δV ′ s′ (ξ ′)ϕV′(ξ′)dξ′, whichcoincideswith (39) in thiscase. Assumeq1>0.Keeping(16)and(18) inmind,weput ξ˜11 := ξ11− tvψ(ξ′)−1v=(φ1(ξ)−1)−111 >0. Bythechangeofvariablesξ= ι(ξ˜11+ tvψ(ξ′)−1v, v, ξ′),wehavedξ=2q1/2dξ˜11dvdξ′. Ontheother hand,weobserve: (x|ξ)= x11(ξ˜11+ tvψ(ξ′)−1v)+2tuv+(x′|ξ′) = x11ξ˜11+x11t(v+x−111 ψ(ξ ′)u)ψ(ξ′)−1(v+x−111 ψ(ξ ′)u)+(x−x−111UtU|ξ′). Thus, the left-handsideof (39)equals:∫ ∞ 0 ∫ R q1 ∫ P∗V′ e−x11ξ˜11e−x11 t(v+x−111 ψ(ξ ′)u)ψ(ξ′)−1(v+x−111 ψ(ξ ′)u)e−(x−x −1 11U tU|ξ′) ×ξ˜s1−1−q1/211 (detψ(ξ′))−1/2δV ′ s′ (ξ ′)ϕV′(ξ′)2q1/2dξ˜11dvdξ′. (40) BytheGaussian integral formula,wehave:∫ R q1 e−x11 t(v+x−111 ψ(ξ ′)u)ψ(ξ′)−1(v+x−111 ψ(ξ ′)u)dv=πq1/2x−q1/211 (detψ(ξ ′))1/2, so that (40)equals: (2π)q1/2x−q1/211 ∫ ∞ 0 e−x11ξ˜11ξ˜s1−1−q1/211 dξ˜11 ∫ P∗V′ e−(x−x −1 11U tU|ξ′)δV ′ s′ (ξ ′)ϕV′(ξ′)dξ′, whichcoincideswith (39)because: ∫∞ 0 e −x11ξ˜11ξ˜s1−1−q1/211 dξ˜11=Γ(s1−q1/2)x−s1+q1/211 . Example 3. LetZV be as in Example 1, and let x ∈ PV and ξ ∈ P∗V be as in (2) and (25), respectively. Then,wehave for s=(s1,s2,s3)∈C3, ΔVs (x)=(x211) s1/2−s2 ∣∣∣∣∣∣∣ x1 0 x4 0 x1 0 x4 0 x2 ∣∣∣∣∣∣∣ s2−s3 (detx)s3 = xs1−s2−s311 ∣∣∣∣∣x1 x4x4 x2 ∣∣∣∣∣ s2−s3 (detx)s3, and: δVs (ξ)=(ξ1− ξ24 ξ2 − ξ 2 5 ξ3 )s1(ξ2− ξ 2 6 ξ3 )s2ξs33 = ∣∣∣∣∣∣∣ ξ1 ξ4 ξ5 ξ4 ξ2 0 ξ5 0 ξ3 ∣∣∣∣∣∣∣ s1 ∣∣∣∣∣ξ2 ξ6ξ6 ξ3 ∣∣∣∣∣ s2 ξ−s12 ξ s3−s1−s2 3 . 245