Page - 434 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 375 AppendixA. ProofsofMonotonicity LemmaA1. β(t)= [( ν−a ν−a+t )ν−a+t( b−ν b−ν−t )b−ν−t] nb−a is strictlydecreasing in t. Proof. We show the equivalent statement that β˜(t) = ln [( ν−a ν−a+t )ν−a+t( b−ν b−ν−t )b−ν−t] is strictly decreasing in t: d dt β˜(t)= d dt (( ln(ν−a)− ln(ν−a+ t))(ν−a+ t)+(ln(b−ν)− ln(b−ν− t))(b−ν− t)) = ln(ν−a)− ln(ν−a+ t)− 1ν−a+t(ν−a+ t)− ln(b−ν)+ ln(b−ν− t)+ 1b−ν−t(b−ν− t) = ln ( b−ν− t b−ν︸ ︷︷ ︸ <1 · ν−a ν−a+ t︸ ︷︷ ︸ <1 ) <0. Hence, β˜(t)andthusβ(t)arestrictlydecreasing in t. LemmaA2. Let t= t(γ,ν,a,b)bethesolutiontotheequationβ(t)=γ.Then,ν+ t is strictly increasing inν. Proof. t is thesolutionof theequation (ν−a+ t)ln ( ν−a ν−a+ t ) +(b−ν− t)ln ( b−ν b−ν− t ) = b−a n lnγ. (A1) The derivatives of the left-hand side of Equation (A1) w.r.t. ν and t exist and are continuous. Furthermore, thederivativew.r.t. tdoesnotvanishforany t∈ (0,b−ν), cf. theproofofLemmaA1, whence the derivative t′ = dtdν exists by the implicit function theorem. When differentiating Equation(A1)withrespect toν, oneobtains (1+ t′)ln ( ν−a ν−a+ t ) +(ν−a+ t) ( 1 ν−a− 1+ t′ ν−a+ t ) −(1+ t′)ln ( b−ν b−ν− t ) +(b−ν− t) ( − 1 b−ν+ 1+ t′ b−ν− t ) =0, orequivalently (1+ t′) [ ln ( ν−a ν−a+ t ) ︸ ︷︷ ︸ <0 −ln ( b−ν b−ν− t ) ︸ ︷︷ ︸ >0 ] = t(a−b) (v−a)(b−v)<0, whence1+ t′= ddν(ν+ t)>0finishes theproof. LemmaA3. The function ξ ( σ̂2 ) = [( 1− σ̂2 1−Vn )1−Vn( σ̂2 Vn )Vn]n is strictlydecreasing in σ̂2∈ [Vn,1]. 434