Page - 103 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 l’esprit à cette opération, est certainementd’unusageplus étenduquecelui où tout est soumisà l’évidence; parceque les occasionsde sedéterminer surdesvraisemblancesouprobabilités, sontplus fréquentesquecelles qui exigentqu’onprocèdepardémonstrations: pourquoinedirions–nouspasque souvent elles tiennentaussi àdesobjetsbeaucoupplus importants? —JosephdeMaistreinL’EspitdeFinesse[221] Le cadavre qui s’acoutre se méconnait et imaginant l’éternité s’en approrie l’illusion . . . C’est pourquoi j’abandonnerai ces frusques et jetant lemasquedemes jours, je fuirai le tempsoù,deconcert avec lesautres, jem’éreinte àme trahir. —EmileCioraninPrécisdedecomposition[222] Conflictsof Interest:Theauthordeclaresnoconflictof interest. AppendixA. Clairaut(-Legendre)EquationofMauriceFréchetAssociatedto“Distinguished Functions”asFundamentalEquationofInformationGeometry BeforeRao [223,224], in 1943,Maurice Fréchet [141]wrote a seminal paper introducingwhat was thencalled theCramer-Raobound. Thispapercontains in factmuchmore that this important discovery. Inparticular,MauriceFréchet introducesmoregeneralnotionsrelative to“distinguished functions”,densitieswithestimator reachingthebound,definedwitha function, solutionofClairaut’s equation. Thesolutions“envelopeof theClairaut’sequation”areequivalent tostandardLegendre transformwithout convexity constraintsbutonly smoothnessassumption. ThisFréchet’s analysis canberevisitedonthebasisof Jean-LouisKoszul’sworksasaseminal foundationof“information geometry”. WewilluseMauriceFréchetnotations, toconsider theestimator: T=H(X1,...,Xn) (A1) andtherandomvariable A(X)= ∂logpθ(X) ∂θ (A2) thatareassociatedto: U=∑ i A(Xi) (A3) Thenormalizingconstraint +∞ −∞ pθ(x)dx=1 implies that: +∞ −∞ ... +∞ −∞ ∏ i pθ(xi)dxi=1 Ifweconsider thederivative if this lastexpressionwithrespect toθ, then +∞ −∞ ... +∞ −∞ [ ∑ i A(xi) ] ∏ i pθ(xi)dxi=0gives :Eθ [U]=0 (A4) Similarly, ifweassumethatEθ [T]= θ, then +∞ −∞ ... +∞ −∞ H(x1,...,xn)∏ i pθ(xi)dxi= θ, andweobtain byderivationwithrespect toθ: E [(T−θ)U]=1 (A5) ButasE [T]= θandE [U]=0,we immediatelydeduce that: E [(T−E [T])(U−E [U])]=1 (A6) FromSchwarz inequality,wecandevelopthe followingrelations: [E(ZT)]2≤E[Z2]E[T2] 1≤E [ (T−E [T])2 ] E [ (U−E [U])2 ] =(σTσU) 2 (A7) 103