Page - 104 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Ubeingthesummationof independentvariables,Bienayméequalitycouldbeapplied: (σU) 2=∑ i [ σA(Xi) ]2 =n(σA) 2 (A8) Fromwhich,Fréchetdeducedthebound, rediscoveredbyCramerandRao2years later: (σT) 2≥ 1 n(σA) 2 (A9) Fréchet [141]observedthat it isaremarkable inequalitywherethesecondmember is independent of thechoiceof the functionHdefiningthe“empiricalvalue”T,where thefirstmembercanbe taken toanyempiricalvalueT=H(X1,...,Xn) subject to theuniqueconditionEθ [T]= θ regardless isθ. Theclassicconditionthat theSchwarzinequalitybecomesanequalityhelpsustodeterminewhen σT reaches its lowerbound 1√nσn . Thepreviousinequalitybecomesanequalityiftherearetwonumbersαandβ (notrandomandnot bothzero )suchthatα(H′−θ)+βU=0,withH′beingaparticular functionamongeligibleH such thatwehaveanequality. Thisequality is rewrittenH′= θ+λ′Uwithλ′beinganon-randomnumber. Ifweuse thepreviousequation, then: E [(T−E [T])(U−E [U])]=1⇒E[(H′−θ)U]=λ′Eθ[U2]=1 (A10) Weobtain: U=∑ i A(Xi)⇒λ′nEθ [ A2 ] =1 (A11) Fromwhichweobtainλ′ andthe formof theassociatedestimatorH′: λ′= 1 nE [A2] ⇒H′= θ+ 1 nE [A2]∑i ∂logpθ(Xi) ∂θ (A12) It is thereforededucedthat theestimator that reaches the terminal isof the form: H′= θ+ ∑ i ∂logpθ(Xi) ∂θ n +∞ −∞ [ ∂pθ(x) ∂θ ]2 dx pθ(x) (A13) withE [H′]= θ+λ′E [U]= θ. H′wouldbeoneof theeligible functions, ifH′wouldbe independentofθ. Indeed, ifweconsider Eθ0 [H ′]= θ0,E [ (H′−θ0)2 ] ≤Eθ0 [ (H−θ0)2 ] ∀H suchthatEθ0 [H]= θ0. H= θ0 satisfies theequationandinequalityshowsthat it isalmostcertainlyequal toθ0. So to lookforθ0,weshouldknowbeforehandθ0. Atthisstage,Fréchet[141]lookedfor“distinguishedfunctions”(“densitésdistinguées”inFrench), asanyprobabilitydensity pθ(x) suchthat the function: h(x)= θ+ ∂logpθ(x) ∂θ +∞ −∞ [ ∂pθ(x) ∂θ ]2 dx pθ(x) (A14) is independent of θ. The objective of Fréchet is then to determine theminimizing function T = H′(X1,...,Xn) that reaches thebound.Wecandeduce frompreviousrelations that: λ(θ) ∂logpθ(x) ∂θ = h(x)−θ (A15) 104