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Joint modeling for mixed-effects quantile regression of longitudinal data with detection limits and covariates measured with error, with application to AIDS studies

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Abstract

It is very common in AIDS studies that response variable (e.g., HIV viral load) may be subject to censoring due to detection limits while covariates (e.g., CD4 cell count) may be measured with error. Failure to take censoring in response variable and measurement errors in covariates into account may introduce substantial bias in estimation and thus lead to unreliable inference. Moreover, with non-normal and/or heteroskedastic data, traditional mean regression models are not robust to tail reactions. In this case, one may find it attractive to estimate extreme causal relationship of covariates to a dependent variable, which can be suitably studied in quantile regression framework. In this paper, we consider joint inference of mixed-effects quantile regression model with right-censored responses and errors in covariates. The inverse censoring probability weighted method and the orthogonal regression method are combined to reduce the biases of estimation caused by censored data and measurement errors. Under some regularity conditions, the consistence and asymptotic normality of estimators are derived. Finally, some simulation studies are implemented and a HIV/AIDS clinical data set is analyzed to to illustrate the proposed procedure.

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Acknowledgements

The authors thank the editors and two reviewers for their constructive comments and valuable suggestions which have greatly improved the paper. The work is jointly supported National Natural Science Foundation of China (No. 11501167), Research Grant Council of the Hong Kong Special Administration Region (No. UGC/FDS14/P01/16), Young Academic Leaders Project of Henan University of Science and Technology (No. 13490008) and China Postdoctoral Science Foundation (No. 2017M610156).

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Correspondence to Manlai Tang.

Appendix

Appendix

The proof of Theorem 1:

According to Cui (1997), Ma and Yin (2011) and Fleming and Harrington (1991), for \((\beta , b)\) in tight set, we have

$$\begin{aligned} Q_{Nm}(\beta , b)= & {} \frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})}\rho _{\tau } \left( \frac{Y_{ij}^{*}-W_{ij}^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}} \right) \\= & {} \frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\rho _{\tau } \left( \frac{Y_{ij}^{*}-W_{ij}^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) \\&+\frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}(G_{0}(Y_{ij}^{*})-\hat{G}(Y_{ij}^{*}))}{\hat{G}(Y_{ij}^{*})G_{0}(Y_{ij}^{*})}\rho _{\tau } \left( \frac{Y_{ij}^{*}-W_{ij}^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) \\= & {} E\left[ E\left\{ \frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} \rho _{\tau }\left( \frac{Y_{ij}^{*}-W_{ij}^{T}\beta -A_{ij} ^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) \Vert W_{ij},Y_{ij},b_{i}\right\} \right] +o_{p}(1)\\= & {} E\left\{ \rho _{\tau }\left( \frac{Y_{ij}-W_{ij}^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) \right\} +o_{p}(1)\\= & {} E\left\{ \rho _{\tau }\left( \frac{\varepsilon _{ij}-U_{ij}^{T}\beta - X_{ij}^{T}(\beta -\beta ^{*})-A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}\right) \right\} +o_{p}(1)\\= & {} E\left\{ \rho _{\tau }\left( \frac{X_{ij}^{T}\beta ^{*}+A_{ij}^{T}b_{i}^{*}+\varepsilon _{ij}-(X_{ij}+U_{ij})^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) \right\} +o_{p}(1)\\= & {} E\left\{ \rho _{\tau }\left( \epsilon _{ij}-\frac{X_{ij}^{T}(\beta -\beta ^{*})+A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}\right) \right\} +o_{p}(1), \end{aligned}$$

where \(\epsilon _{ij}\) and \(\frac{\varepsilon _{ij}-U_{ij}^{T}\beta }{\sqrt{1+||\beta ||^{2}}}\) are distributed as the same spherically symmetric distribution and hence have the same mean and variance. Since \(\epsilon _{ij}\), \(X_{ij}\)and \(b_{i}\) are independent, we can conclude that \(Q_{Nm}(u)\) converges to its expectation; i,e.,

$$\begin{aligned} Q_{Nm}(\beta , b)\overset{P}{\rightarrow }E\left\{ \rho _{\tau }\left( e_{ij}-\frac{X_{ij}^{T}(\beta -\beta ^{*})+A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}\right) \right\} . \end{aligned}$$

From assumption, for any \(\tau \) we have \(E\Big \{\rho _{\tau }\Big (\epsilon _{ij}-\frac{X_{ij}^{T}(\beta -\beta ^{*})+A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}\Big )\Big \}\) has unique solution. Therefore, we have \(\frac{X_{ij}^{T}(\beta -\beta ^{*})+A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}=0\), namely, \(\beta =\beta ^{*}, b_{i}=b_{i}^{*}.\)

Let \(\hat{\beta }\) and \(\hat{b}\) be the minimum points such that \(\Big \{\frac{X_{ij}^{T}(\tilde{\beta }-\beta ^{*})}{\sqrt{1+||\tilde{\beta }||^{2}}},\frac{A_{ij}^{T}(\tilde{b}_{i}-b_{i}^{*})}{\sqrt{1+||\tilde{\beta }||^{2}}}\Big \}\) converges to \((\beta _{c},b_{c})\). For any \(c_{\tau }\), we have

$$\begin{aligned} E\left\{ \rho _{\tau }\left( \epsilon _{ij}-\frac{X_{ij}^{T}(\beta -\beta ^{*})+A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}\right) \right\} \le E\left\{ \rho _{\tau }\left( e_{ij}-c_{\tau }\right) \right\} . \end{aligned}$$

In particular, setting \(c_{\tau }=0\) yields \(\beta _{c}=0\) and \(b_{c}=0\), and the proof is completed. \(\square \)

The proof of Theorem 2:

Let

$$\begin{aligned} u=\left( \begin{array}{c} u^{(1)} \\ u^{(2)}\\ \end{array} \right) =\left( \begin{array}{c} \sqrt{Nm}(\beta -\beta ^{*}) \\ \sqrt{m}(b-b^{*}) \\ \end{array} \right) ,v_{ij}=\frac{X_{ij}^{T}u^{(1)}}{\sqrt{N}}+A_{ij}^{T}u_{i}^{(2)}. \end{aligned}$$

It is easy to see that the proposed estimator \(\hat{u}\) or \((\hat{\beta }, \hat{b})\) is the minimum point of the following objective function

$$\begin{aligned}&L_{Nm}(\hat{G},u)\\&\quad =\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})} \left[ \rho _{\tau }\left( \frac{Y_{ij}^{*}-W_{ij}^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) -\rho _{\tau }\left( \frac{Y_{ij}^{*}-W_{ij}^{T}\beta ^{*}-A_{ij}^{T}b_{i}^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}\right) \right] \\&\quad =\sum _{i=1}^{N}\sum _{j=1}^{m} \frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})}\left[ \rho _{\tau }\left( \frac{X_{ij}^{T}\beta ^{*}+A_{ij}^{T}b_{i}^{*}+\varepsilon _{ij}-(X_{ij}+U_{ij})^{T}\beta -A_{ij}^{T}b_{i}}{\sqrt{1+||\beta ||^{2}}}\right) \right. \\&\qquad \left. -\rho _{\tau }\left( \frac{X_{ij}^{T}\beta ^{*} +A_{ij}^{T}b_{i}^{*}+\varepsilon _{ij}- (X_{ij}+U_{ij})^{T}\beta ^{*}-A_{ij}^{T}b_{i}^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}\right) \right] \\&\quad =\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})}\left[ \rho _{\tau }\left( \frac{\varepsilon _{ij} -U_{ij}^{T}\beta }{\sqrt{1+||\beta ||^{2}}}-\frac{X_{ij}^{T}(\beta -\beta ^{*})+A_{ij}^{T}(b_{i}-b_{i}^{*})}{\sqrt{1+||\beta ||^{2}}}\right) \right. \\&\qquad \left. -\rho _{\tau }\left( \frac{\varepsilon _{ij}-U_{ij}^{T} \beta ^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}\right) \right] \\&\quad =\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})}\left[ \rho _{\tau }\left( \frac{\varepsilon _{ij}-U_{ij}^{T}\beta }{\sqrt{1+||\beta ||^{2}}}-\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}\right) -\rho _{\tau }\left( \frac{\varepsilon _{ij}-U_{ij}^{T}\beta ^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}\right) \right] \\&\quad =\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})}\left[ \rho _{\tau }\left( \epsilon _{ij}-\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}\right) -\rho _{\tau }(\epsilon _{ij}^{*})\right] \\&\quad =\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{\hat{G}(Y_{ij}^{*})}\left[ \rho _{\tau }\left( \epsilon _{ij}^{*}-(\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij})\right) -\rho _{\tau }(\epsilon _{ij}^{*})\right] , \end{aligned}$$

where \(h_{ij}=\epsilon _{ij}-\epsilon _{ij}^{*}, \epsilon _{ij}\triangleq \frac{\varepsilon _{ij}-U_{ij}^{T}\beta }{\sqrt{1+||\beta ||^{2}}}, \epsilon _{ij}^{*}\triangleq \frac{\varepsilon _{ij}-U_{ij}^{T}\beta ^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}.\)

According to the unit spherically symmetric distribution of \((\varepsilon ,U_{ij}^{T})^{T}\), we know that \(\epsilon _{ij}\) and \(\epsilon _{ij}^{*}\) are unit spherically symmetric distribution. Hence, \(h_{ij}=\epsilon _{ij}-\epsilon _{ij}^{*}\) is also symmetrically distributed. According to Theorem 3 in Cui (1997), for \(h_{ij}\) applying the Taylor expansion on \(\frac{1}{\sqrt{1+||\beta ||^{2}}}\) and \(\frac{\beta _{j}}{\sqrt{1+||\beta ||^{2}}}\) at \(\beta ^{*}\) yields

$$\begin{aligned} \frac{1}{\sqrt{1+||\beta ||^{2}}}= & {} \frac{1}{\sqrt{1+||\beta ^{*}||^{2}}}-\frac{(\beta ^{*})^{T}(\beta -\beta ^{*})}{(1+||\beta ^{*}||^{2})^{3/2}}- \frac{1}{2}(\beta -\beta ^{*})^{T}Q(\bar{\beta })(\beta -\beta ^{*}),\\ \frac{\beta _{l}}{\sqrt{1+||\beta ||^{2}}}= & {} \frac{\beta _{j}^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}+ \left( \frac{e_{l}}{\sqrt{1+||\beta ^{*}||^{2}}}- \frac{\beta _{l}^{*}\beta ^{*}}{(1+||\beta ^{*}||^{2})^{3/2}}\right) ^{T}(\beta -\beta ^{*})\\&- (\beta -\beta ^{*})^{T}Q_{l}(\bar{\beta }_{l})(\beta -\beta ^{*}), l=1,2,\ldots ,p, \end{aligned}$$

where

$$\begin{aligned}&Q(\beta )=\frac{I_{p}}{(1+||\beta ||^{2})^{3/2}}-\frac{3\beta \beta ^{T}}{(1+||\beta ||^{2})^{5/2}},\\&Q_{l}(\beta )=\frac{\beta _{l}I_{p}+e_{l}\beta ^{T}+\beta e_{l}^{T}}{(1+||\beta ||^{2})^{3/2}}-\frac{3\beta _{l}\beta \beta ^{T}}{(1+||\beta ||^{2})^{5/2}},\\&\gamma =\beta ^{*}+\theta (\beta -\beta ^{*}), \gamma _{l}=\beta ^{*}+\theta _{l}(\beta -\beta ^{*}), 0\le \theta ,\theta _{l}\le 1,\\&\bar{\beta }=\beta ^{*}+\bar{\theta }(\beta -\beta ^{*}), \bar{\beta }_{l}=\beta ^{*}+\bar{\theta }_{l}(\beta -\beta ^{*}), 0\le \bar{\theta },\bar{\theta }_{l}\le 1,\\&e_{l}=(0,\ldots ,0,l,0,\ldots ,0), 1\le l\le p. \end{aligned}$$

\(h_{ij}\) can be decomposed as follows,

$$\begin{aligned} h_{ij}= & {} \frac{\varepsilon _{ij}-U_{ij}^{T}\beta }{\sqrt{1+||\beta ||^{2}}}-\frac{\varepsilon _{ij}-U_{ij}^{T}\beta ^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}\\= & {} -\left( \frac{U_{ij}}{\sqrt{1+||\beta ^{*}||^{2}}}+ \frac{(\varepsilon _{ij}-U_{ij}\beta ^{*})\beta ^{*}}{(1+||\beta ^{*}||^{2})^{3/2}}\right) ^{T}(\beta -\beta ^{*}) -\frac{1}{2}(\beta -\beta ^{*})^{T}\cdot \\&[\varepsilon _{ij}Q(\bar{\beta })-2\sum _{l=1}^{p}U_{ijl}Q_{l}(\bar{\beta }_{l})]\cdot (\beta -\beta ^{*})\\= & {} -\left( \frac{U_{ij}}{\sqrt{1+||\beta ^{*}||^{2}}}+ \frac{\epsilon _{ij}^{*}\beta ^{*}}{1+||\beta ^{*}||^{2}}\right) ^{T}(\beta -\beta ^{*})-\frac{1}{2}(\beta -\beta ^{*})^{T} \cdot \\&[\varepsilon _{ij}Q(\bar{\beta })-2\sum _{l=1}^{p}U_{ijl}Q_{l}(\bar{\beta }_{l})]\cdot (\beta -\beta ^{*})\\= & {} -\frac{\Psi _{ij}^{T}}{\sqrt{1+||\beta ^{*}||^{2}}}(\beta -\beta ^{*}){-}\frac{1}{2}(\beta {-}\beta ^{*})^{T}[\varepsilon _{ij}Q(\bar{\beta }){-}2\sum _{l=1}^{p}U_{ijl}Q_{l}(\bar{\beta }_{l})](\beta -\beta ^{*})\\= & {} -\frac{\Psi _{ij}^{T}}{\sqrt{1+||\beta ^{*}||^{2}}}(\beta -\beta ^{*})+o_{p}(||\beta -\beta ^{*}||)\\= & {} -\frac{\Psi _{ij}^{T}}{\sqrt{Nm}\sqrt{1+||\beta ^{*}||^{2}}}u^{(1)}+\frac{1}{\sqrt{Nm}}o_{p}(||u^{(1)}||),\end{aligned}$$

where \(\Psi _{ij}=U_{ij}+\frac{\epsilon _{ij}^{*}\beta ^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}.\)

Hence, \(L_{Nm}(\hat{G},u)\) can be decomposed as follows

$$\begin{aligned} L_{Nm}(\hat{G},u)= & {} V_{Nm}^{(1)}+V_{Nm}^{(2)}-V_{Nm}^{(3)}+o_{p}(1), \end{aligned}$$

where

$$\begin{aligned}&V_{Nm}^{(1)}=\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} \left[ \rho _{\tau }\left( \epsilon _{ij}^{*} -(\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij})\right) -\rho _{\tau }(\epsilon _{ij}^{*})\right] ,\\&V_{Nm}^{(2)}=\sum _{i=1}^{N}\sum _{j=1}^{m}\delta _{ij} \cdot \rho _{\tau }\left( \epsilon _{ij}^{*}-\left( \frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}\right) \right) \left( \frac{1}{\hat{G}(Y_{ij}^{*})}-\frac{1}{G_{0}(Y_{ij}^{*})}\right) , and\\&V_{Nm}^{(3)}=\sum _{i=1}^{N}\sum _{j=1}^{m}\delta _{ij} \cdot \rho _{\tau }(\epsilon _{ij}^{*})\left( \frac{1}{\hat{G}(Y_{ij}^{*})}-\frac{1}{G_{0}(Y_{ij}^{*})}\right) .\end{aligned}$$

\(\square \)

Applying Knight equation in Knight (1998) gives

$$\begin{aligned} \rho _{\tau }(x-y)-\rho _{\tau }(x)=y(I(x<0)-\tau )+\int _{0}^{y}[I(x\le t)-I(x\le 0)]dt,~x\ne 0. \end{aligned}$$

Let \((\beta ,b)\) be in the local area of \((\beta ^{*},b^{*})\). \(V_{Nm}^{(1)}\) can be decomposed as follows

$$\begin{aligned} V_{Nm}^{(1)}= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}(\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij})(I(\epsilon _{ij}^{*}<0)-\tau )\\&+\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\int _{0}^{(\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij})}[I(\epsilon _{ij}^{*}\le t)-I(\epsilon _{ij}^{*}\le 0)]dt\\= & {} \frac{1}{\sqrt{m}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\left( \frac{X_{ij}^{T}u^{(1)}}{\sqrt{N}}+A_{ij}^{T}u_{i}^{(2)}\right) \frac{(I(\epsilon _{ij}^{*}<0)-\tau )}{\sqrt{1+||\beta ||^{2}}}\\&-\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}h_{ij}(I(\epsilon _{ij}^{*}<0)-\tau )\\&+\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\int _{0}^{\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}}[I(\epsilon _{ij}^{*}\le t)-I(\epsilon _{ij}^{*}\le 0)]dt\\\triangleq & {} W_{Nm,1}^{T}u^{(1)}+W_{Nm,2}^{T}u^{(2)}+T_{Nm}+B_{Nm}\\\triangleq & {} W_{Nm}^{T}u+T_{Nm}+B_{Nm},~~W_{Nm}^{T}=(W_{Nm,1}^{T},W_{Nm,2}^{T}), \end{aligned}$$

where

$$\begin{aligned} W_{Nm,1}= & {} \frac{1}{\sqrt{Nm}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}X_{ij}\frac{(I(\epsilon _{ij}^{*}<0)-\tau )}{\sqrt{1+||\beta ||^{2}}}\\= & {} \frac{1}{\sqrt{Nm}\sqrt{1+||\beta ^{*}||^{2}}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}X_{ij}(I(\epsilon _{ij}^{*}<0)-\tau )+o_{p}(1),\\ W_{Nm,2}= & {} \frac{1}{\sqrt{m}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} A_{ij}\frac{(I(\epsilon _{ij}^{*}<0)-\tau )}{\sqrt{1+||\beta ||^{2}}}\\= & {} \frac{1}{\sqrt{m}\sqrt{1+||\beta ^{*}||^{2}}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} A_{ij}(I(\epsilon _{ij}^{*}<0)-\tau )+o_{p}(1),\\ T_{Nm}= & {} -\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}h_{ij}(I(\epsilon _{ij}^{*}<0)-\tau )\\= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}[\frac{\Psi _{ij}^{T}}{\sqrt{Nm}\sqrt{1+||\beta ^{*}||^{2}}}u^{(1)}\\&+\frac{1}{\sqrt{Nm}}o_{p}(||u^{(1)}||)](I(\epsilon _{ij}^{*}<0)-\tau )\\= & {} \left( \frac{1}{\sqrt{Nm}\sqrt{1+||\beta ^{*}||^{2}}}\sum _{i=1}^{N} \sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} \Psi _{ij}^{T}(I(\epsilon _{ij}^{*}<0)-\tau )\right. \\&\left. +\frac{1}{\sqrt{Nm}}\sum _{i=1}^{N}\sum _{j=1}^{m} \frac{\delta _{ij}(I(\epsilon _{ij}^{*}<0)-\tau )}{G_{0}(Y_{ij}^{*})}o_{p}(1)\right) u^{(1)}\\\triangleq & {} \left( W_{Nm,3}+\frac{1}{\sqrt{Nm}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}(I(\epsilon _{ij}^{*}<0)-\tau )}{G_{0}(Y_{ij}^{*})}o_{p}(1)\right) ^{T}u^{(1)}, \end{aligned}$$

where

$$\begin{aligned} W_{Nm,3}= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\frac{\Psi _{ij}}{\sqrt{Nm}\sqrt{1+||\beta ^{*}||^{2}}}(I(\epsilon _{ij}^{*}<0)-\tau ),\\ B_{Nm}= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\int _{0}^{\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}} [I(\epsilon _{ij}^{*}\le t)-I(\epsilon _{ij}^{*}\le 0)]dt. \end{aligned}$$

The second equalities in \(W_{Nm,1}\) and \(W_{Nm,2}\) can be obtained using the above Taylor expansions and the Law of Large Number.

From the assumption of the spherically symmetric distribution of \(\epsilon _{ij}^{*}\) and \(\varepsilon _{ij}\), we can obtain \(E[I(\epsilon _{ij}^{*}<0)-\tau ]=0\). Under the regular condition C.3, by Lindeberg-Feller Central Limit Theorem and Cramer–Vold theorem, we know that \(W_{Nm,1}\) and\( W_{Nm,2}\) converge to \(W_{1}\) and \(W_{2 }\) in distribution, respectively. Here, \(W_{1}\) and \(W_{2}\) are normally distributed with zero mean.

For \(T_{Nm}\), we know from Central Limit Theorem that \(W_{Nm,3}\) converges to \(W_{3}\) in distribution. From Central Limit Theroem and Slutsky Theorem, the second term in \(T_{Nm}\)

$$\begin{aligned} \frac{1}{\sqrt{Nm}}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}(I(\epsilon _{ij}^{*}<0)-\tau )}{G_{0}(Y_{ij}^{*})}o_{p}(1) \end{aligned}$$

converges to 0 in probability. Hence, we can conclude that the term inside the blanket of the last equality in \(T_{Nm}\) also converges to \(W_{3}\).

By Cramer–Vold theorem, we have

$$\begin{aligned} W_{Nm,1}^{T}u^{(1)}+W_{Nm,2}^{T}u^{(2)}+T_{Nm}{\mathop {\rightarrow }\limits ^{d}}(W_{1}+W_{3})^{T}u^{(1)}+W_{2}^{T}u^{(2)}=W^{T}u, \end{aligned}$$

where \(W=((W_{1}+W_{3})^{T},W_{2}^{T})^{T}\).

For \(B_{Nm}\), we have decomposition: \(B_{Nm}=E(B_{Nm})+[B_{Nm}-E(B_{Nm})],\) where

$$\begin{aligned} E[B_{Nm}]= & {} E\left[ \sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\int _{0}^{\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}} [I(\epsilon _{ij}^{*}\le t)-I(\epsilon _{ij}^{*}\le 0)]dt\right] \\= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}E\left\{ E\left[ \frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})}\int _{0}^{\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}} [I(\epsilon _{ij}^{*}\le t)-I(\epsilon _{ij}^{*}\le 0)]dt\Big |Y_{ij} \right] \right\} \\= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}E\left\{ \int _{0}^{\frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}}[F_{ij}(t|X_{ij},A_{ij},b_{i}) -F_{ij}(0|X_{ij},A_{ij},b_{i})]dt\Big |h_{ij}\right\} \\= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}E\left\{ \int _{0}^{\frac{v_{ij}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}}s\cdot f_{ij}(0|X_{ij}, A_{ij},b_{i})ds\Big |h_{ij}\right\} +o(1)\\= & {} \frac{1}{2}\sum _{i=1}^{N}\sum _{j=1}^{m}f_{ij}(0|X_{ij},A_{ij},b_{i})\cdot E \left( \frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}\right) ^{2}+o(1)\\= & {} \frac{1}{2m(1+||\beta ^{*}||^{2})}\sum _{i=1}^{N}\sum _{j=1}^{m}f_{ij}(0|X_{ij},A_{ij},b_{i})\\&\cdot E\left( \frac{(X_{ij}^{T}u^{(1)})^{2}}{N}+ \frac{2(X_{ij}^{T}u^{(1)})\cdot (A_{ij}^{T}u_{i}^{(2)}) }{\sqrt{N}}\right. \\&\left. +\,(A_{ij}^{T}u_{i}^{(2)})^{2}+2\frac{(X_{ij}^{T}u^{(1)})/\sqrt{N}+A_{ij}^{T}u_{i}^{(2)}}{\sqrt{N}}\Psi _{ij}^{T}u^{(1)} +\frac{(\Psi _{ij}^{T}u^{(1)})^{2}}{N}\right) +o(1)\\= & {} \frac{1}{2m(1+||\beta ^{*}||^{2})}\left( \frac{(u^{(1)})^{T}\cdot E[(X+\Psi )^{T}\Omega (X+\Psi )]u^{(1)}}{N}\right. \\&\left. +\,2\frac{(u^{(1)})^{T}(X+E\Psi )^{T}\Omega Au^{(2)}}{\sqrt{N}}+(u^{(2)})^{T}A^{T}\Omega Au^{(2)}\right) +o(1)\\\rightarrow & {} \frac{1}{2}u^{T}Du, \end{aligned}$$

where

$$\begin{aligned} D=\lim _{N,m\rightarrow \infty }\frac{1}{m}\cdot \left( \begin{array}{cc} E[(X+\Psi )^{T}\Omega (X+\Psi )]/N &{}\quad A^{T}\Omega (X+E\Psi )/\sqrt{N} \\ (X+E\Psi )^{T}\Omega A/\sqrt{N} &{} A^{T}\Omega A \\ \end{array} \right) . \end{aligned}$$

The computation of D involves the calculations of the expectation and second moment of \(\Psi \), which can be readily computed by noting that \(\Psi _{ij}=U_{ij}+\frac{\epsilon _{ij}^{*}\beta ^{*}}{\sqrt{1+||\beta ^{*}||^{2}}}\) and the assumption of spherically symmetric distribution.

From conditions of C.4 and C.5, we have

$$\begin{aligned} Var[B_{Nm}]\le & {} \frac{4}{\zeta _{0}\sqrt{1+||\beta ||^{2}}} E[B_{Nm}]\cdot \max _{\le i\le N,1\le j\le m}\left| \frac{X_{ij}^{T}u^{(1)}}{\sqrt{Nm}}+\frac{A_{ij}^{T}u_{i}^{(2)}}{\sqrt{m}}\right| \\\le & {} \frac{4}{\zeta _{0}\sqrt{1+||\beta ||^{2}}} E[B_{Nm}]\\&\cdot \left( \max _{\le i\le N,1\le j\le m}\left| \frac{X_{ij}^{T}u^{(1)}}{\sqrt{Nm}}\left| +\max _{\le i\le N,1\le j\le m}\right| \frac{A_{ij}^{T}u_{i}^{(2)}}{\sqrt{m}}\right| \right) \\\rightarrow & {} 0. \end{aligned}$$

Therefore, we have \(B_{Nm}{\mathop {\rightarrow }\limits ^{d}}\frac{1}{2}u^{T}Du\). By Cramer–Vold theorem, we have

$$\begin{aligned} R_{Nm}^{(1)}=W^{T}u+\frac{1}{2}u^{T}Du+o_{p}(1). \end{aligned}$$

It is easy to see from Shows et al. (2010) and Tang et al. (2012), a martingale integral representation which is similar to Gill (1980) can be given as follows

$$\begin{aligned} \sqrt{Nm}\left( \frac{1}{G(Y_{ij}^{*})}-\frac{1}{G_{0}(Y_{ij}^{*})}\right)= & {} -\sqrt{Nm}\frac{(G(Y_{ij}^{*})-G_{0}(Y_{ij}^{*}))}{G^{2}_{0}(Y_{ij}^{*})}+o_{p}(1)\\= & {} \frac{1}{G_{0}(Y_{ij}^{*})}\frac{1}{\sqrt{Nm}}\sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}I(Y_{ij}^{*}\ge t)\frac{dM_{rs}^{C}(t)}{H(t)}\\&+\,\, o_{p}(1), \end{aligned}$$

where \(H(t)=\lim _{N,m\rightarrow \infty }\frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}I(Y_{ij}^{*}\ge t)\), \(M_{ij}^{C}(t)=(1-\delta _{ij})I(Y_{ij}^{*}\le t)-\int _{0}^{c}I(Y_{ij}^{*}\ge t)d\Lambda _{C}(t)\), and \(\Lambda _{C}(t)\) is the cumulative hazard function of censoring variable C.

Using the above expression, we have

$$\begin{aligned} V_{Nm}^{(2)}= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}\delta _{ij} \cdot \rho _{\tau }\left( \epsilon _{ij}^{*}-\left( \frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}\right) \right) \left( \frac{1}{\hat{G}(Y_{ij}^{*})}- \frac{1}{G_{0}(Y_{ij}^{*})}\right) \\= & {} \frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} \cdot \rho _{\tau }\left( \epsilon _{ij}^{*}-\left( \frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}\right) \right) \\&\cdot \sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}I(Y_{ij}^{*}\ge t)\frac{dM_{rs}^{C}(t)}{H(t)}+o_{p}(1),\\ V_{Nm}^{(3)}= & {} \sum _{i=1}^{N}\sum _{j=1}^{m}\delta _{ij} \cdot \rho _{\tau }(\epsilon _{ij}^{*})\left( \frac{1}{\hat{G}(Y_{ij}^{*})}-\frac{1}{G_{0}(Y_{ij}^{*})}\right) \\= & {} \frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} \cdot \rho _{\tau }(\epsilon _{ij}^{*})\cdot \sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}I(Y_{ij}^{*}\ge t)\frac{dM_{rs}^{C}(t)}{H(t)}+o_{p}(1). \end{aligned}$$

Furthermore, we have

$$\begin{aligned}&V_{Nm}^{(2)}-V_{Nm}^{(3)}\\&\quad =\frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}}{G_{0}(Y_{ij}^{*})} \cdot \left[ \rho _{\tau }\left( \epsilon _{ij}^{*}-\left( \frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ||^{2}}}-h_{ij}\right) \right) -\rho _{\tau }(\epsilon _{ij}^{*})\right] \\&\qquad \cdot \sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}I(Y_{ij}^{*}\ge t)\frac{dM_{rs}^{C}(t)}{H(t)}+o_{p}(1)\\&\quad =\frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m}\frac{\delta _{ij}I(\epsilon _{ij}^{*}\le 0)-\tau ]}{G_{0}(Y_{ij}^{*})}\\&\qquad \cdot \left( \frac{v_{ij}/\sqrt{m}}{\sqrt{1+||\beta ^{*}||^{2}}}-h_{ij})\cdot \sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}I(Y_{ij}^{*}\ge t\right) \frac{dM_{rl}^{C}(t)}{H(t)}+o_{p}(1)\\&\quad =\frac{1}{Nm}\sum _{i=1}^{N}\sum _{j=1}^{m} \frac{\frac{(X_{ij}+\Psi _{ij})^{T}u^{(1)}}{\sqrt{N}}+A_{ij}^{T}u_{i}^{(2)}}{\sqrt{m}\sqrt{1+||\beta ^{*}||^{2}}}\\&\qquad \cdot \sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}\frac{\delta _{ij}[I(\epsilon _{ij}^{*}\le 0)-\tau ]I(Y_{ij}^{*}\ge t)}{G_{0}(Y_{ij}^{*})}\frac{dM_{rs}^{C}(t)}{H(t)}+o_{p}(1). \end{aligned}$$

Let

$$\begin{aligned} \mu _{Nm}^{(ij)}= & {} \frac{(X_{ij}+\Psi _{ij})^{T}}{\sqrt{N}}\\&\cdot \frac{1}{Nm\sqrt{1+||\beta ^{*}||^{2}}}\sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}\frac{\delta _{ij}[I(\epsilon _{ij}^{*}\le 0)-\tau ]I(Y_{ij}^{*}\ge t)}{G_{0}(Y_{ij}^{*})}\frac{dM_{rs}^{C}(t)}{H(t)},\\ \xi _{Nm}^{(ij)}= & {} A_{ij}^{T} \cdot \frac{1}{Nm\sqrt{1+||\beta ^{*}||^{2}}}\sum _{r=1}^{N}\sum _{s=1}^{m}\int _{0}^{c}\frac{\delta _{ij}[I(\epsilon _{ij}^{*}\le 0)-\tau ]I(Y_{ij}^{*}\ge t)}{G_{0}(Y_{ij}^{*})}\frac{dM_{rs}^{C}(t)}{H(t)}. \end{aligned}$$

According to the Martingale Central Limit Theorem (see, Fleming and Harrington 1991) and Cramer–Vold theorem, we have

$$\begin{aligned} \frac{1}{\sqrt{m}}\sum _{i=1}^{N}\sum _{j=1}^{m}\mu _{Nm}^{(ij)}u^{(1)}&{\mathop {\rightarrow }\limits ^{d}}W_{4}^{T}u^{(1)},~~ \frac{1}{\sqrt{m}}\sum _{i=1}^{N}\sum _{j=1}^{m}\xi _{Nm}^{(ij)}u_{i}^{(2)}&{\mathop {\rightarrow }\limits ^{d}}W_{5}^{T}u^{(2)}. \end{aligned}$$

where \(W_{4}\) and \(W_{5}\) are normal random vectors with zero mean. Also, we have

$$\begin{aligned} L_{Nm}(\hat{G},u){\mathop {\rightarrow }\limits ^{d}}(W+W')^{T}u+\frac{1}{2}u^{T}Du,~\text {where}~W'=(W_{4}^{T},W_{5}^{T})^{T}. \end{aligned}$$

By Knight (1998) and Koenker (2005), we can readily show that

$$\begin{aligned} u{\mathop {\rightarrow }\limits ^{d}}N(0,D^{-1}\Sigma D^{-1}). \end{aligned}$$

where \(\Sigma \) is the covariance matrix of \((W+W')\).

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Tian, Y., Tang, M. & Tian, M. Joint modeling for mixed-effects quantile regression of longitudinal data with detection limits and covariates measured with error, with application to AIDS studies. Comput Stat 33, 1563–1587 (2018). https://doi.org/10.1007/s00180-018-0812-0

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