Page - 277 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 is satisfied forallα∈ [α0,α1], then thederivativeofκ(α) exists at anyα∈ (α0,α1), and isgivenby ∂κ ∂α (α)=−1 2 ∫ T[ϕ −1(q)−ϕ−1(p)]ϕ′(cα)dμ∫ T ϕ ′(cα)u0dμ , (16) where cα= 1−α2 ϕ −1(p)+ 1+α2 ϕ −1(q)+κ(α)u0. Proof. Forα∈ (α0,α1)andκ>0,define g(α,κ)= ∫ T ϕ (1−α 2 ϕ−1(p)+ 1+α 2 ϕ−1(q)+κu0 ) dμ. The functionκ(α) isdefinedimplicitlybyg(α,κ(α))=1. Ifweshowthat (i) the functiong(α,κ) is continuous inaneighborhoodof (α,κ(α)), (ii) thepartialderivatives ∂g∂α and ∂g ∂κ existandarecontinuousat (α,κ(α)), (iii) and ∂g∂κ(α,κ(α))>0, thenbythe ImplicitFunctionTheoremκ(α) isdifferentiableatα∈ (α0,α1), and ∂κ ∂α (α)=−(∂g/∂α)(α,κ(α)) (∂g/∂κ)(α,κ(α)) . (17) Webeginbyverifying that g(α,κ) is continuous. Forfixedα∈ (α0,α1)andκ> 0, setκ0= 2κ. DenotingA={t∈T : ϕ−1(q(t))>ϕ−1(p(t))},wecanwrite ϕ (1−β 2 ϕ−1(p)+ 1+β 2 ϕ−1(q)+λu0 ) ≤ϕ ( ϕ−1(p)+ 1+β 2 [ϕ−1(q)−ϕ−1(p)]+κ0u0 ) ≤ϕ ( ϕ−1(p)+ 1+α1 2 [ϕ−1(q)−ϕ−1(p)]+κ0u0 ) 1A +ϕ ( ϕ−1(p)+ 1+α0 2 [ϕ−1(q)−ϕ−1(p)]+κ0u0 ) 1T\A, (18) foreveryβ∈ (α0,α1)andλ∈ (0,κ0). Because the functionontheright-handsideof (18) is integrable, wecanapply theDominatedConvergenceTheoremtoconcludethat lim (β,λ)→(α,κ) g(β,λ)= g(α,κ). Now, wewill show that the derivative of g(α,κ)with respect to α exists and is continuous. Consider thedifference g(γ,λ)−g(β,λ) γ−β = ∫ T 1 γ−β [ ϕ ( cβ+ γ−β 2 [ϕ−1(q)−ϕ−1(p)]+λu0 ) −ϕ(cβ+λu0) ] dμ, (19) where cβ= 1−β 2 ϕ −1(p)+ 1+β2 ϕ −1(q). Representby fβ,γ,λ the function inside the integral sign in(19). Forfixedα∈ (α0,α1)andκ>0,denoteα0=(α0+α)/2,α1=(α+α1)/2,andκ0=2κ. Becauseϕ(·) is convexandincreasing, it followsthat |fβ,γ,λ|≤ fα1,α1,κ01A− fα0,α0,κ01T\A=: f, forallβ,γ∈ (α0,α1)andλ∈ (0,κ0), whereA={t∈T : ϕ−1(q(t))>ϕ−1(p(t))}.Observingthat f is integrable,wecanusetheDominated ConvergenceTheoremtoget lim γ→β ∫ T fβ,γ,λdμ= ∫ T ( lim γ→β fβ,γ,λ ) dμ, 277