89 We can verify the conditions of Theorem 8.2 to show that the MLE is consistent.First,we know that Qo()is uniquely maximized at 0o since we can show that 0o is identified.Does there exist 00o so that p(x;0)=p(x;00)?If so,then it must be the case that (x-0)2 =(x-00)2 for all x.This can only happen if =00.Thus, 0o is identified.By assumption,we know that e is compact.To show continuity of Qo()and uniform convergence in probability of Q(;Xn)to Qo(0),we appeal to the conditions of Lemma 8.3.We have to show that logp(x;0)is continuous in 0 for 0e and all x e Y.This function clearly satisfies this continuity condition. Finally,we have to show that there exists a function d(x)such that |log p(x;f)l≤d(x)for all0∈Θandx∈Y and Eo[d(X)】<o
89 We can verify the conditions of Theorem 8.2 to show that the MLE is consistent. First, we know that Q0(θ) is uniquely maximized at θ0 since we can show that θ0 is identified. Does there exist θ = θ0 so that p(x; θ) = p(x; θ0)? If so, then it must be the case that (x − θ)2 = (x − θ0)2 for all x. This can only happen if θ = θ0. Thus, θ0 is identified. By assumption, we know that Θ is compact. To show continuity of Q0(θ) and uniform convergence in probability of Q(θ; Xn) to Q0(θ), we appeal to the conditions of Lemma 8.3. We have to show that log p(x; θ) is continuous in θ for θ ∈ Θ and all x ∈ X . This function clearly satisfies this continuity condition. Finally, we have to show that there exists a function d(x) such that | log p(x; θ)| ≤ d(x) for all θ ∈ Θ and x ∈ X and Eθ0 [d(X)] < ∞
90 Note that there exist positive constants C1,C2>1 and C3 so that Ilogp(x;0)=I-log-log(1+(x-0)2) logπ+log(1+(x-0)2) C1+log(C2+C3x2)=d(x) It remains to show that Eold(X)]<oo.Note that Eold(x)C:+log(Ca+Ca00 1 -dx -a+bgc+ca+日- 1 -dx oCC =( 1 d 60 = G+人aec+ce+ao9a十西c+ a(C+C+u 1 00
90 Note that there exist positive constants C1, C2 > 1 and C3 so that | log p(x; θ)| = | − log π − log(1 + (x − θ)2)| = log π + log(1 + (x − θ)2) ≤ C1 + log(C2 + C3x2) = d(x) It remains to show that Eθ0 [d(X)] < ∞. Note that Eθ0 [d(X)] = ∞−∞{C1 + log(C2 + C3x2)} 1 π(1 + (x − θ0)2) dx = C1 + ∞−∞ log(C2 + C3x2) 1 π(1 + (x − θ0)2)dx = C1 + ∞−∞ log(C2 + C3(x + θ0)2) 1 π(1 + x2)dx = C1 + −θ0 −∞ log(C2 + C3(x + θ0)2) 1 π(1 + x2) dx + ∞−θ0 log(C2 + C3(x + θ0)2) 1 π(1 + x2) dx