Page - 105 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Butasλ(θ)>0,wecanconsider 1 λ(θ) as the secondderivativeof a functionΦ(θ) suchthat: ∂logpθ(x) ∂θ = ∂2Φ(θ) ∂θ2 [h(x)−θ] (A16) Fromwhichwededuce that: (x)= logpθ(x)− ∂Φ(θ)∂θ [h(x)−θ]−Φ(θ) (A17) Isan independentquantityofθ.Adistinguished functionwillbe thengivenby: pθ(x)= e ∂Φ(θ) ∂θ [h(x)−θ]+Φ(θ)+ (x) (A18) With thenormalizingconstraint +∞ −∞ pθ(x)dx=1. These twoconditionsaresufficient. Indeed, reciprocally, let three functionsΦ(θ),h(x)and (x) thatwehave, forany θ : +∞ −∞ e ∂Φ(θ) ∂θ [h(x)−θ]+Φ(θ)+ (x)dx=1 (A19) Thenthe function isdistinguished: θ+ ∂logpθ(x) ∂θ +∞ −∞ [ ∂pθ(x) ∂θ ]2 dx pθ(x) = θ+λ(x) ∂2Φ(θ) ∂θ2 [h(x)−θ] (A20) Ifλ(x) ∂2Φ(θ) ∂θ2 =1, when 1 λ(x) = +∞ −∞ [ ∂logpθ(x) ∂θ ]2 pθ(x)dx=(σA) 2 (A21) Thefunction is reducedtoh(x)andthen isnotdependentofθ. Wehavethenthe followingrelation: 1 λ(x) = +∞ −∞ ( ∂2Φ(θ) ∂θ2 )2 [h(x)−θ]2e∂Φ(θ)∂θ (h(x)−θ)+Φ(θ)+ (x)dx (A22) Therelation isvalid foranyθ,wecanderiveprefiousequationwithrespectwithθ: +∞ −∞ e ∂Φ(θ) ∂θ (h(x)−θ)+Φ(θ)+ (x) ( ∂2Φ(θ) ∂θ2 ) [h(x)−θ]dx=0 (A23) Wecandivideby ∂2Φ(θ) ∂θ2 because itdoesnotdependonx. Ifwederiveagainwithrespect toθ,wewillhave: +∞ −∞ e ∂Φ(θ) ∂θ (h(x)−θ)+Φ(θ)+ (x) ( ∂2Φ(θ) ∂θ2 ) [h(x)−θ]2dx= +∞ −∞ e ∂Φ(θ) ∂θ (h(x)−θ)+Φ(θ)+ (x)dx=1 (A24) 105