(A.24), we have
J2(Qθ)=O(T-1h-m). (A.33)
Next, we show that the result of Step 3 in the proof of Theorem (i) holds under the
alternative hypothesis. Since F(Qθ (Xj )∣ zj ) - θ ≠ Ofor j = t, s under the alternative
hypothesis, we have
J3(Qθ)-J3(Qθ-CT)
1 ɪ ɪ 1 ( Z.-Z I
=∑ ∑ mK∖-lγ~^l×{F(Qθ(x)|zt)-θ}{F(Qθ(x°)|Zs)-θ}
T (T 1) t=1 s ≠ th L h J
1TT1(ZZA
--1— ∑ ∑ J- K l^-ʌ I
t(T -1) ⅛ ⅛ hm L h J
× {F(Qθ(xt)-Ct∣z1 )-θ}{F(Qθ(xs)-Qk)-θ}
1T
= 7 ∑ ! F(Qθ(X, ) I z, ) - θi!F(Qθ (Xs ) I zs ) - θ}f ■ (Z, )
T t=1
1T
- 7 ∑ {F ( Q ( Xt ) - Ct∣z, ) - θ}{ F ( Q. ( x ) - Cτ∣z, ) - θ} fz ( z, ). (A.34)
T t=1
By taking a Taylor expansion of Fyj, (Qθ (Xj ) - Ct | zj ) around Qθ (zj ) for j = t, s, we
have
J3(Qθ)-J3(Qθ-CT)
1T
= -∑ {F(Qe(Xt)∖zt)-θ}CτS(Qe(Xs))f,(z,)
T t=1
1T
+7∑ CtS(Qθ(X, )){F(Qθ(Xs ) I z,) -θifz(z■,)
T t=1
-ɪ∑ CTS(Qθ(X. ))S(Qθ(Xs)).f■ (z, ). (A.35)
T t=1
We further take Taylor expansion of Fy∣z (Qθ (Xj ) | zj) around Qθ (Zj ) for j = t, s and have
J3(Qθ)-J3(Qθ-CT)
20