Page - 242 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 4.Γ-TypeIntegralFormulas Forann×nmatrixA=(Aij)and1≤m≤n,wedenotebyA[m] theupper-leftm×m submatrix (Aij)i,j≤mofA. PutMk :=∑ki=1nk (k=1,. . . ,r). For s=(s1, . . . ,sr)∈Cr,wedefinefunctionsΔVs on PV andδVs onP∗V respectivelyby: ΔVs (x) :=(detx[M1])s1/n1 r ∏ k=2 ( detx[Mk] detx[Mk−1] )sk/nk (26) =(detx)sr/nr r−1 ∏ k=1 (detx[Mk])sk/nk−sk−1/nk−1 (x∈PV), δVs (ξ) := r ∏ k=1 (φk(ξ) −1)−sk11 (27) =∏ qk=0 ξ sk kk∏ qk>0 (ξkk− tvkψk(ξ)−1vk)sk (ξ∈P∗V). Recall (22) for thesecondequalityof (27). For a=(a1, . . . ,ar)∈Rr>0, letDadenote thediagonalmatrixdefinedby: Da := ⎛⎜⎜⎜⎜⎝ a1In1 a2In2 ... arInr ⎞⎟⎟⎟⎟⎠∈GL(n,R). Then, the linearmapZV x →DaxDa∈ZV preservesbothPV andP∗V, andwehave: ΔVs (DaxDa)=( r ∏ k=1 a2skk )Δ V s (x) (x∈PV), (28) δVs (DaξDa)=( r ∏ k=1 a2skk )δ V s (ξ) (ξ∈PV). (29) Assumeq1>0. ForB∈W,wedenotebyτB the linear transformonZV givenby: τBx := ( In1 B In−n1 )( x11In1 tU U x′ )( In1 tB In−n1 ) = ( x11In1 tU+x11tB U+x11B x′+UtB+BtU+x11BtB ) , wherex∈ZV isas in (3). Indeed, since: UtB+BtU=(U+B)t(U+B)−UtU−BtB∈ZV′, thematrixτBxbelongs toZV. Clearly,τBpreservesPV, andwehave: ΔVs (τBx)=ΔVs (x) (x∈PV). (30) Theformula (5) is rewrittenas: τ−x−111U(x)= ( x11In1 x′−x−111UtU ) , 242