Page - 107 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Wecanrewrite theestimatoras: H′(X1,...,Xn)= 1 n [h(X1)+ ...+h(Xn)] (A32) andcompute theassociatedempiricalvalue: t=H′(x1,...,xn)= 1 n∑i h(xi)= θ+λ(θ)∑ i ∂logpθ(xi) ∂θ Ifwetakeθ= t,wehaveasλ(θ)>0: ∑ i ∂logpt(xi) ∂t =0 (A33) When pθ(x) is adistinguished function, the empirical value tof θ corresponding to a sample x1,...,xn is a root of previous equation in t. This equation has a root and only onewhenX is a distinguishedvariable. Indeed,aswehave: pθ(x)= e ∂Φ(θ) ∂θ [h(x)−θ]+Φ(θ)+ (x) (A34) ∑ i ∂logpt(xi) ∂t = ∂2Φ(t) ∂t2 ⎡⎣∑i h(xi) n − t ⎤⎦with ∂2Φ(t) ∂t2 >0 (A35) Wecanthenrecover theuniqueroot: t= ∑ i h(xi) n . This function T ≡ H′(X1,...,Xn) = 1n∑ i h(Xi) can have an arbitrary form, that is a sumof functionsofeachonlyoneof thequantitiesandit iseventhearithmeticaverageofNvaluesofasame auxiliaryrandomvariableY= h(X). Thedispersion isgivenby: (σTn) 2= 1 n(σA) 2 = 1 n +∞ −∞ [ ∂pθ(x) ∂θ ]2 dx pθ(x) = 1 n ∂2Φ(θ) ∂θ2 (A36) andTn followstheprobabilitydensity: pθ(t)= √ n 1 σA √ 2π e −n(t−θ)2 2·σ2A with (σA) 2= ∂2Φ(θ) ∂θ2 (A37) ClairautEquationandLegendreTransform Wehave just observed that Fréchet shows thatdistinguished functionsdependona function Φ(θ), solutionof theClairautequation: Φ(θ)= θ · ∂Φ(θ) ∂θ −Ψ ( ∂Φ(θ) ∂θ ) (A38) OrgivenbytheLegendre transform: Φ= θ ·s−Ψ(s) andθ= ∂Ψ(s) ∂s (A39) Fréchetalsoobservedthat this functionΦ(θ)couldberewritten: Φ(θ)=−log +∞ −∞ es·h(x)+ (x)dx+θ ·swheres isgivenimplicitlyby +∞ −∞ es·h(x)+ (x) [h(x)−θ]dx=0. 107