7 Lemma 8.1:If 0o is identified and Eo[l logp(X;0)]<oo for all 0∈Θ，Qo(0)is uniquely maximized at0=fo. Proof:By Jensen's inequality,we know that for any strictly convex function g(),E[g(Y)]>g(E[Y]).Take g(y)=-log(y).So, for0卡0o, Eoo[-log( >-品》 Note that 厂密-r= So,Ea[-log(〗>0or Qo(0o)=Eoo[logp(X;00)]>E0o [logp(X;0)]=Qo(0) This inequality holds for all 000.7 Lemma 8.1: If θ0 is identified and Eθ0 [| log p(X; θ)|] < ∞ for all θ ∈ Θ, Q0(θ) is uniquely maximized at θ = θ0. Proof: By Jensen’s inequality, we know that for any strictly convex function g(·), E[g(Y )] > g(E[Y ]). Take g(y) = − log(y). So, for θ = θ0, Eθ0 [− log( p(X; θ) p(X; θ0))] > − log(Eθ0 [ p(X; θ) p(X; θ0)]) Note that Eθ0 [ p(X; θ) p(X; θ0)] =  p(x; θ) p(x; θ0)p(x; θ0)dµ(x) =  p(x; θ)=1 So, Eθ0 [− log( p(X;θ) p(X;θ0) )] > 0 or Q0(θ0) = Eθ0 [log p(X; θ0)] > Eθ0 [log p(X; θ)] = Q0(θ) This inequality holds for all θ = θ0.