Page - 435 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 375 Proof. Weshowtheequivalentstatement thatn−1lnξ ( σ̂2 ) is strictlydecreasing in σ̂2 : d dσ̂2 [ n−1lnξ ( σ̂2 )] = d dσ̂2 [ (1−Vn) ( ln(1− σ̂2)− ln(1−Vn) ) +Vn ( ln(σ̂2)− ln(Vn) )] =−1−Vn 1− σ̂2︸ ︷︷ ︸ >1 + Vn σ̂2︸︷︷︸ <1 <0. LemmaA4. Letw=w(γ,ρ2)be the solutionof the equation [( 1+ w ρ2 )−ρ2−w( 1−w )w−1] n1+ρ2 =γ. Then,w is increasing inρ2. Proof. w is thesolutionof theequation ρ2+w 1+ρ2 ln ( 1+ w ρ2 ) + 1−w 1+ρ2 ln ( 1−w)=−lnγ n . (A2) The derivatives of the left-hand side of Equation (A2) w.r.t. ρ2 and w exist and are continuous. Furthermore, thederivativew.r.t.wdoesnotvanish foranyw∈ (0,1): thisderivative is 1 1+ρ2 [ ln ( 1+ w ρ2 ) + ρ2+w ρ2 ( 1+ w ρ2 )− ln(1−w)−1]= 1 1+ρ2 [ ln ( 1+ w ρ2 ) − ln(1−w) ] , vanishing ifandonly if1+ w ρ2 =1−w, i.e., if andonly ifw(1+ 1 ρ2 ) =0,whichdoesnothappenfor w,ρ2>0.Now, thederivativew′= dwdρ2 existsbythe implicit functiontheorem.Whendifferentiating Equation(A2)withrespect toρ2,oneobtains (1+w′)(1+ρ2)−(ρ2+w) (1+ρ2)2 ln ( 1+ w ρ2 ) + ρ2+w 1+ρ2 · w′ ρ2 − w ρ4 1+ w ρ2︸ ︷︷ ︸ w′ρ2−w ρ2(1+ρ2) −w ′(1+ρ2)+(1−w) (1+ρ2)2 ln(1−w)− w ′ 1+ρ2 =0, orequivalently w′ [ ln ( 1+ w ρ2 ) − ln(1−w)︸ ︷︷ ︸ >0 ] = w ρ2 − 1−w 1+ρ2 ln ( ρ2+w ρ2(1−w) ) . Hence,w′ ≥0 ifandonly if w ρ2 ≥ 1−w1+ρ2 ln ( ρ2+w ρ2(1−w) ) ,whichholdssince ln ( ρ2+w ρ2(1−w) ) = ln ( 1+w(1+ρ 2) ρ2(1−w) )≤ w ρ2 1+ρ2 1−w ,finishingtheproof. References 1. Mardia,K.V.DirectionalStatistics;AcademicPress: London,UK,1972. 435