Based on (
2.7), we have
$$\begin{aligned} 0 =&\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}\bigl(Z_{i}^{T}\breve{\alpha}_{n}-t_{0} \bigr) \left ( \textstyle\begin{array}{@{}c@{}}1\\ Z_{i}^{T}\breve{\alpha}_{n}-t_{0} \end{array}\displaystyle \right ) \\ &{}\cdot\bigl[Y_{i}^{*}-V_{i}^{T}\breve{ \beta}_{n}-\breve{g}_{n}(t_{0})-\breve {g}'_{n}(t_{0}) \bigl(Z_{i}^{T} \breve{\alpha}_{n}-t_{0}\bigr)\bigr]. \end{aligned}$$
Taking only the top equation into account, using a Taylor expansion, and calculating directly, we obtain
$$\begin{aligned} &\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}\bigl(Z_{i}^{T}\alpha-t_{0}\bigr) \bigl[\breve {g}_{n}(t_{0})-g(t_{0})\bigr] \\ &\quad =\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}\bigl(Z_{i}^{T}\alpha-t_{0}\bigr) \bigl[(1-\delta_{i})V_{i}^{T}(\hat{ \beta}_{n}-\beta)+(1-\delta_{i}) \bigl(\hat {g}_{n} \bigl(Z_{i}^{T}\alpha\bigr)-g\bigl(Z_{i}^{T} \alpha\bigr)\bigr) \\ &\qquad {}+\delta_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr)\bigr]-(\breve{\beta}_{n}-\beta)^{T} \frac {1}{n}\sum_{i=1}^{n} K_{h_{4}}\bigl(Z_{i}^{T}\alpha-t_{0} \bigr)V_{i} \\ &\qquad {}-(\breve{\alpha}_{n}-\alpha)^{T} \frac{1}{n}\sum _{i=1}^{n} K_{h_{4}} \bigl(Z_{i}^{T}\alpha-t_{0}\bigr)Z_{i} g'(t_{0}) +o_{p}\biggl(\frac{1}{\sqrt{n}} \biggr)+O_{p}\bigl(h_{4}^{2}\bigr). \end{aligned}$$
(5.15)
Dividing all terms in (
5.15) by
\(\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})\), we have
$$\begin{aligned} \breve{g}_{n}(t_{0})-g(t_{0}) =& \frac{\frac{1}{n}\sum_{i=1}^{n} \delta_{i} K_{h_{4}}(Z_{i}^{T}\alpha -t_{0})(\varepsilon_{i}-U_{i}^{T}\beta)}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} \\ &{}+(\hat{\beta}_{n}-\beta)^{T} \frac{\frac{1}{n}\sum_{i=1}^{n} (1-\delta_{i}) K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})V_{i}}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} \\ &{}+\bigl(\hat{g}_{n}(t_{0})-g(t_{0})\bigr) \frac{\frac{1}{n}\sum_{i=1}^{n} (1-\delta_{i}) K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} \\ &{}-(\breve{\beta}_{n}-\beta)^{T} \frac{\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})V_{i}}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} \\ &{}-(\breve{\alpha}_{n}-\alpha)^{T} \frac{\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})Z_{i} g'(t_{0})}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} +o_{p}\biggl(\frac{1}{\sqrt{n}}\biggr)+O_{p} \bigl(h_{4}^{2}\bigr). \end{aligned}$$
Note that
$$\begin{aligned}& \frac{\frac{1}{n}\sum_{i=1}^{n} (1-\delta_{i}) K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})V_{i}}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} = E\bigl[(1-\delta) X|Z^{T} \alpha=t_{0} \bigr]\bigl(1+o_{p}(1)\bigr), \\& \frac{\frac{1}{n}\sum_{i=1}^{n} (1-\delta_{i}) K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} = 1-E\bigl(\delta|Z^{T} \alpha=t_{0} \bigr) \bigl(1+o_{p}(1)\bigr), \\& \frac{\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})V_{i}}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} = E\bigl(X|Z^{T} \alpha=t_{0}\bigr) \bigl(1+o_{p}(1)\bigr), \end{aligned}$$
and
$$\begin{aligned} \frac{\frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})Z_{i} g'(t_{0})}{ \frac{1}{n}\sum_{i=1}^{n} K_{h_{4}}(Z_{i}^{T}\alpha-t_{0})} =E\bigl( Z g'\bigl(Z^{T}\alpha \bigr)|Z^{T} \alpha=t_{0}\bigr) \bigl(1+o_{p}(1) \bigr). \end{aligned}$$
Thus, equation (
5.14) follows.
Second, we give the proof of Theorem
3.2. From (
2.9),
\(\breve{\alpha}_{n}\) is the solution of
$$\frac{1}{n}\sum_{i=1}^{n} \bigl[Y_{i}^{*}-V_{i}^{T}\breve{ \beta}_{n}-\breve {g}_{n}\bigl(Z_{i}^{T} \breve{\alpha}_{n}\bigr)\bigr] \cdot\breve{g}'_{n} \bigl(Z_{i}^{T}\breve{\alpha}_{n} \bigr)Z_{i}=0, $$
it can be rewritten as
$$\begin{aligned} &\frac{1}{n}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \bigl\{ \bigl[Y_{i}^{*}-V_{i}^{T}\beta-g \bigl(Z_{i}^{T}\alpha\bigr)\bigr]-\bigl[\breve{g}_{n} \bigl(Z_{i}^{T}\breve{\alpha }_{n}\bigr)-g \bigl(Z_{i}^{T}\alpha\bigr)\bigr] \\ &\quad {}-V_{i}^{T}(\breve{\beta}_{n}-\beta)\bigr\} \cdot \bigl(1+o_{p}(1)\bigr)=0. \end{aligned}$$
(5.16)
Because of the Taylor expansion and the continuity of
\(g'(\cdot)\), we can obtain
$$\begin{aligned} &\breve{g}_{n}\bigl(Z_{i}^{T} \breve{\alpha}_{n}\bigr)-g\bigl(Z_{i}^{T}\alpha \bigr) \\ &\quad = \breve{g}_{n}\bigl(Z_{i}^{T}\alpha \bigr)+g'\bigl(Z_{i}^{T}\alpha\bigr) \bigl(Z_{i}^{T}\breve{\alpha }_{n}-Z_{i}^{T} \alpha\bigr) -g\bigl(Z_{i}^{T}\alpha\bigr)+o_{p} \biggl(\frac{1}{\sqrt{n}}\biggr). \end{aligned}$$
(5.17)
By (
5.17), (
5.16) can be written as
$$\begin{aligned} &\frac{1}{n}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \bigl\{ \delta_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr)+(1-\delta_{i})V_{i}^{T}(\hat{\beta }_{n}-\beta) \\ &\quad {}+(1-\delta_{i})\bigl[\hat{g}_{n}\bigl(Z_{i}^{T} \alpha\bigr)-g\bigl(Z_{i}^{T}\alpha\bigr)\bigr] -\bigl[ \breve{g}_{n}\bigl(Z_{i}^{T}\alpha\bigr)-g \bigl(Z_{i}^{T}\alpha\bigr)\bigr] -V_{i}^{T}( \breve{\beta}_{n}-\beta) \\ &\quad {}-g'\bigl(Z_{i}^{T}\alpha \bigr)Z_{i}^{T}(\breve{\alpha}_{n}-\alpha)\bigr\} \bigl(1+o_{p}(1)\bigr)=0. \end{aligned}$$
Applying (
5.14) to the equation, it is easy to obtain
$$\begin{aligned} &\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i} g'\bigl(Z_{i}^{T}\alpha \bigr)Z_{i}\bigl(\varepsilon _{i}-U_{i}^{T} \beta\bigr) \\ &\qquad {}-\frac{1}{\sqrt{n}}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \cdot\frac{\frac{1}{n}\sum_{j=1}^{n} \delta_{j} K_{h_{4}}(Z_{j}^{T} \alpha -Z_{i}^{T}\alpha)(\varepsilon_{j}-U_{j}^{T}\beta)}{ \frac{1}{n}\sum_{j=1}^{n} K_{h_{4}}(Z_{j}^{T} \alpha-Z_{i}^{T}\alpha)} \\ &\qquad {}-\frac{1}{\sqrt{n}}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \bigl[\hat {g}_{n}\bigl(Z_{i}^{T}\alpha\bigr)-g \bigl(Z_{i}^{T}\alpha\bigr)\bigr] \cdot\bigl[ \delta_{i}-E\bigl(\delta|Z^{T} \alpha=Z_{i}^{T} \alpha\bigr)\bigr] \\ &\qquad {}+(\hat{\beta}_{n}-\beta)^{T}\frac{1}{\sqrt{n}}\sum _{i=1}^{n} g' \bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \bigl[(1- \delta_{i})V_{i}-E\bigl((1-\delta)V|Z^{T} \alpha=Z_{i}^{T}\alpha\bigr)\bigr] \\ &\quad =\frac{1}{\sqrt{n}}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \left ( \textstyle\begin{array}{@{}c@{}} \widetilde{\widetilde{Z}}_{i} g'(Z_{i}^{T}\alpha)\\ \widetilde{\widetilde{X}}_{i}+U_{i} \end{array}\displaystyle \right )^{T} \left ( \textstyle\begin{array}{@{}c@{}} \breve{\alpha}_{n}-\alpha\\ \breve{\beta}_{n}-\beta \end{array}\displaystyle \right )+o_{p}(1). \end{aligned}$$
(5.18)
Note that the second term of the left-hand side of (
5.18) is
$$\begin{aligned} \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr)E\bigl[Z g'\bigl(Z^{T}\alpha \bigr)|Z^{T} \alpha=Z_{i}^{T}\alpha \bigr]+o_{p}(1). \end{aligned}$$
Then the first two terms of the left-hand side of (
5.18) are as follows:
$$\begin{aligned} &\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i} g'\bigl(Z_{i}^{T}\alpha \bigr)Z_{i}\bigl(\varepsilon _{i}-U_{i}^{T} \beta\bigr){} {} {} {} {} {} {} {} {} \\ &\qquad {}-\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i}\bigl(\varepsilon _{i}-U_{i}^{T} \beta\bigr)E\bigl[Z g'\bigl(Z^{T}\alpha \bigr)|Z^{T} \alpha=Z_{i}^{T}\alpha\bigr] \\ &\quad =\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta \bigr)\widetilde{\widetilde{Z}}_{i} g' \bigl(Z_{i}^{T}\alpha\bigr). \end{aligned}$$
(5.19)
Applying (
5.1) to the third term of the left-hand side of (
5.18), it follows that
$$\begin{aligned} &\frac{1}{\sqrt{n}}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \bigl[\delta_{i}-E\bigl(\delta |Z^{T} \alpha=Z_{i}^{T} \alpha\bigr)\bigr] \cdot\bigl[\hat{g}_{n}\bigl(Z_{i}^{T} \alpha\bigr)-g\bigl(Z_{i}^{T}\alpha\bigr)\bigr] \\ &\quad =\frac{1}{\sqrt{n}}\sum_{i=1}^{n} g'\bigl(Z_{i}^{T}\alpha\bigr)Z_{i} \bigl[\delta_{i}-E\bigl(\delta |Z^{T} \alpha=Z_{i}^{T} \alpha\bigr)\bigr] \\ &\qquad {}\cdot\Biggl\{ \frac{1}{n} \frac{1}{f(Z_{i}^{T}\alpha)\mu(Z_{i}^{T}\alpha)} \sum _{j=1}^{n} \delta_{j} K_{h_{2}} \bigl(Z_{j}^{T} \alpha-Z_{i}^{T}\alpha \bigr) \bigl(\varepsilon _{j}-U_{j}^{T}\beta\bigr) \Biggr\} \\ &\qquad {}-(\hat{\beta}_{n}-\beta)^{T} \frac{1}{\sqrt{n}}\sum _{i=1}^{n} g' \bigl(Z_{i}^{T}\alpha \bigr)Z_{i}\bigl[ \delta_{i}-E\bigl(\delta|Z^{T} \alpha=Z_{i}^{T} \alpha\bigr)\bigr] \frac{E(\delta X|Z^{T} \alpha=Z_{i}^{T}\alpha)}{E(\delta|Z^{T} \alpha =Z_{i}^{T}\alpha)} \\ &\qquad {}-(\hat{\alpha}_{n}-\alpha)^{T} \frac{1}{\sqrt{n}}\sum _{i=1}^{n} g' \bigl(Z_{i}^{T}\alpha\bigr)Z_{i}\bigl[ \delta_{i}-E\bigl(\delta|Z^{T} \alpha=Z_{i}^{T} \alpha\bigr)\bigr] \\ &\qquad {}\times \frac{E(\delta Z g'(Z^{T}\alpha)|Z^{T} \alpha=Z_{i}^{T}\alpha)}{E(\delta|Z^{T} \alpha=Z_{i}^{T}\alpha)}+o_{p}(1) =J_{1}-J_{2}-J_{3}+o_{p}(1). \end{aligned}$$
(5.20)
Similar to the second term of the left-hand side of (
5.18),
$$\begin{aligned} J_{1} =&\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \delta_{i}\bigl( \varepsilon_{i}-U_{i}^{T}\beta \bigr) \\ &{}\times \biggl\{ \frac{E[\delta Z g'(Z^{T}\alpha)|Z_{i}^{T}\alpha]}{E(\delta|Z_{i}^{T}\alpha)} -\frac{E[E(\delta|Z^{T}\alpha)Z g'(Z^{T}\alpha)|Z_{i}^{T}\alpha]}{E(\delta |Z_{i}^{T}\alpha)}\biggr\} . \end{aligned}$$
(5.21)
Also, we have
$$\begin{aligned} J_{2}=\sqrt{n}(\hat{\beta}_{n}- \beta)^{T} E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha \bigr)\bigr] \frac{E(\delta X|Z^{T} \alpha)}{E(\delta|Z^{T} \alpha)}g'\bigl(Z^{T}\alpha\bigr)Z \biggr\} +o_{p}(1). \end{aligned}$$
(5.22)
Combining with Lemma
5.3, we have
$$\begin{aligned} J_{3} =&\Biggl[\Gamma_{\widetilde{Z}}^{-1} \Biggl\{ \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i} g'\bigl(Z_{i}^{T}\alpha \bigr)\widetilde {Z}_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr) -\sqrt{n}\Gamma_{\widetilde{Z}\widetilde{X}}(\hat{\beta}_{n}-\beta) \Biggr\} \Biggr]^{T} \\ &{}\cdot E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha\bigr)\bigr] \frac{E[\delta Z g'(Z^{T}\alpha)|Z^{T} \alpha]}{E(\delta|Z^{T} \alpha )}g'\bigl(Z^{T}\alpha\bigr)Z\biggr\} +o_{p}(1). \end{aligned}$$
(5.23)
The last term of the left-hand side of (
5.18) is
$$\begin{aligned} \sqrt{n}(\hat{\beta}_{n}-\beta)^{T} E\bigl\{ \bigl[(1-\delta)X-E\bigl((1-\delta)X|Z^{T}\alpha \bigr) \bigr]g'\bigl(Z^{T}\alpha\bigr)Z\bigr\} +o_{p}(1). \end{aligned}$$
(5.24)
Through a direct calculation, the first term of the right-hand side of (
5.18) is
$$\begin{aligned} \sqrt{n}\Sigma_{\widetilde{\widetilde{Z}}}(\breve{\alpha}_{n}- \alpha)+o_{p}(1), \end{aligned}$$
(5.25)
where
$$\begin{aligned} \Sigma_{\widetilde{\widetilde{Z}}}=E\bigl\{ \bigl[\widetilde{\widetilde{Z}} g'\bigl(Z^{T}\alpha\bigr)\bigr]^{\otimes2}\bigr\} . \end{aligned}$$
(5.26)
The last term of the right-hand side of (
5.18) is
$$\begin{aligned} \sqrt{n}\Sigma_{\widetilde{\widetilde{Z}}\widetilde{\widetilde {X}}}(\breve{\beta}_{n}- \beta)+o_{p}(1), \end{aligned}$$
(5.27)
where
$$\begin{aligned} \Sigma_{\widetilde{\widetilde{Z}}\widetilde{\widetilde{X}}}=E\bigl\{ \widetilde{\widetilde{Z}}g' \bigl(Z^{T}\alpha\bigr)\widetilde{\widetilde{X}}^{T}\bigr\} . \end{aligned}$$
(5.28)
Combining (
5.19)-(
5.25), and (
5.27), and using Theorem
3.1, (
5.18) becomes
$$\begin{aligned} \sqrt{n}\Sigma_{\widetilde{\widetilde{Z}}}(\breve{\alpha}_{n}- \alpha) =& \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr) g'\bigl(Z_{i}^{T}\alpha\bigr) \widetilde{\widetilde{Z}_{i}} \\ &{}-\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta \bigr)g'\bigl(Z^{T}\alpha\bigr) \frac{E[(\delta-E(\delta|Z^{T}\alpha) )Z|Z_{i}^{T}\alpha]}{E(\delta |Z_{i}^{T}\alpha)} \\ &{}+\sqrt{n}(\hat{\beta}_{n}-\beta)^{T} E\biggl\{ \bigl[ \delta-E\bigl(\delta|Z^{T} \alpha\bigr)\bigr] \frac{E(\delta X|Z^{T} \alpha)}{E(\delta|Z^{T} \alpha)}g' \bigl(Z^{T}\alpha\bigr)Z\biggr\} \\ &{}+ E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha\bigr)\bigr] \frac{E[\delta Z g'(Z^{T}\alpha)|Z^{T} \alpha]}{E(\delta|Z^{T} \alpha )}g'\bigl(Z^{T}\alpha\bigr)Z\biggr\} ^{T} \\ & {}\cdot \Biggl[\Gamma_{\widetilde{Z}}^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^{n} \delta_{i} g'\bigl(Z_{i}^{T}\alpha\bigr) \widetilde{Z}_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr)\Biggr] \\ &{}- E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha\bigr)\bigr] \frac{E[\delta Z g'(Z^{T}\alpha)|Z^{T} \alpha]}{E(\delta|Z^{T} \alpha )}g'\bigl(Z^{T}\alpha\bigr)Z\biggr\} ^{T} \\ &{} \cdot \bigl[\Gamma_{\widetilde{Z}}^{-1}\sqrt{n}\Gamma_{\widetilde{Z}\widetilde {X}}( \hat{\beta}_{n}-\beta)\bigr] \\ & {}+\sqrt{n}(\hat{\beta}_{n}-\beta)^{T} E\bigl\{ \bigl[(1- \delta)X-E\bigl((1-\delta)X|Z^{T}\alpha\bigr)\bigr]Z g' \bigl(Z^{T}\alpha\bigr)\bigr\} \\ & {}-\Sigma_{\widetilde{\widetilde{Z}}\widetilde{\widetilde{X}}}\Sigma _{\widetilde{\widetilde{X}}}^{-1} \bigl( \Gamma_{\widetilde{X}}+\Sigma_{1}-\Sigma_{2} \Gamma_{\widetilde {Z}}^{-1}\Gamma_{\widetilde{Z}\widetilde{X}}\bigr)\sqrt{n}(\hat{\beta }_{n}-\beta) \\ &{} +\Sigma_{\widetilde{\widetilde{Z}}\widetilde{\widetilde{X}}}\Sigma _{\widetilde{\widetilde{X}}}^{-1} \Sigma_{2}\Gamma_{\widetilde{Z}}^{-1} \cdot\sqrt{n} \frac{1}{n}\sum_{i=1}^{n} \delta_{i} g'\bigl(Z_{i}^{T}\alpha \bigr)\widetilde{Z}_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta \bigr)+o_{p}(1) \\ =&F_{1}-F_{2}+F_{3}+F_{4}-F_{5}+F_{6}-F_{7}+F_{8}+o_{p}(1). \end{aligned}$$
(5.29)
Through a direct calculation,
$$\begin{aligned} Q =&F_{1}-F_{2}+F_{4}+F_{8} \\ =& \biggl\{ 1+ E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha\bigr) \bigr] \frac{E[\delta Z g'(Z^{T}\alpha)|Z^{T} \alpha]}{E(\delta|Z^{T} \alpha )}g'\bigl(Z^{T}\alpha\bigr)Z\biggr\} ^{T}\Gamma_{\widetilde{Z}}^{-1} \\ &{}+\Sigma_{\widetilde{\widetilde{Z}}\widetilde{\widetilde{X}}}\Sigma _{\widetilde{\widetilde{X}}}^{-1} \Sigma_{2}\Gamma_{\widetilde{Z}}^{-1}\biggr\} \cdot \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \delta_{i} g'\bigl(Z_{i}^{T}\alpha \bigr)\widetilde{Z}_{i}\bigl(\varepsilon_{i}-U_{i}^{T} \beta\bigr). \end{aligned}$$
(5.30)
Combining with Lemma
5.2, we have
$$\begin{aligned} P =&F_{3}-F_{5}+F_{6}-F_{7} \\ =&\biggl[ E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha\bigr) \bigr] \frac{E(\delta X|Z^{T} \alpha)}{E(\delta|Z^{T} \alpha)}g'\bigl(Z^{T}\alpha\bigr)Z\biggr\} ^{T} \\ &{}-E\biggl\{ \bigl[\delta-E\bigl(\delta|Z^{T} \alpha\bigr)\bigr] \frac{E[\delta Z g'(Z^{T}\alpha)|Z^{T} \alpha]}{E(\delta|Z^{T} \alpha )}g'\bigl(Z^{T}\alpha\bigr)Z\biggr\} ^{T} \Gamma_{\widetilde{Z}}^{-1}\Gamma_{\widetilde{Z}\widetilde{X}} \\ &{}+E\bigl\{ \bigl[(1-\delta)X-E\bigl((1-\delta)X|Z^{T}\alpha\bigr) \bigr]Z g'\bigl(Z^{T}\alpha\bigr)\bigr\} ^{T} \\ &{}-\Sigma_{\widetilde{\widetilde{Z}}\widetilde{\widetilde{X}}}\Sigma _{\widetilde{\widetilde{X}}}^{-1} \bigl( \Gamma_{\widetilde{X}}+\Sigma_{1}-\Sigma_{2} \Gamma_{\widetilde {Z}}^{-1}\Gamma_{\widetilde{Z}\widetilde{X}}\bigr)\biggr] \Gamma_{\widetilde {X}}^{-1} \\ &{}\cdot\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \bigl\{ \delta_{i} \bigl[\widetilde{X}_{i} \bigl( \varepsilon_{i}-U_{i}^{T} \beta\bigr) +U_{i} \varepsilon_{i}-\bigl(U_{i} U_{i}^{T}-\Sigma_{uu}\bigr)\beta\bigr]\bigr\} +o_{p}(1). \end{aligned}$$
(5.31)