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A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 345 ✬ ✫ Rao-Cramérjeva ocena Naj bo f gostotna ali verjetnostna funkcija slučajne spremenljivke X in naj bo odvisna ˇse od parametra ζ, tako da je f(x; ζ) njena vrednost v točki x. Zdruˇzeno gostotno ali verjetnostno funkcijo slučajnega vzorca (X1, X2, X3, . . . , Xn) označimo z L in ji pravimo funkcija verjetja (tudi zanesljivosti, angl. likelihood) L(x1, x2, x3, . . . , xn; ζ) = f(x1; ζ)f(x2; ζ)f(x3; ζ) · · · f(xn; ζ) Velja (*): ... L(x1, x2, . . . , xn; ζ)dx1dx2 . . . dxn = 1. L(X1, X2, X3, . . . , Xn) je funkcija vzorca – torej slučajna spremenljivka. Privzemimo, da je funkcija L vsaj dvakrat zvezno odvedljiva po ζ na nekem intervalu I in naj na tem intervalu tudi integral odvoda L po ζ enakomerno konvergira. Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 346 ✬ ✫ . . . Rao-Cramérjeva ocena Odvajajmo enakost (*) po ζ in upoˇstevajmo ... ∂ ln L kar lahko tolmačimo kot E ∂ζ ∂ ln L ∂ζ = 1 L ∂L ∂ζ ∂ ln L ∂ζ Ldx1dx2 . . . dxn = 0 = 0 . pa dobimo Naj bo sedaj C nepristranska cenilka parametra ζ, torej EC = ζ, oziroma zapisano z integrali ... CLdx1dx2 . . . dxn = ζ. Ker C ni odvisna od ζ, dobimo z odvajanjem po ζ: ... ∂ ln L C ∂ζ Ldx1dx2 . . . dxn = 1 kar pomeni E(C ∂ ln L ∂ζ ) = 1 . Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
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✬ ✫ Ljubljana, 6. oktober 2008
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A. Juriˇsić in V. Batagelj: Verje
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