Abstract
This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.
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Notes
In the simulations below, we tried as starting values the following alternative specifications: \(\left[ \beta _{\tau }^{0T},a_{\tau }^{0T},b_{\tau }^{0T}\right] ^{{\mathrm{T}}}=\left[ \widehat{\beta }_{q\tau }^{{T}},\widehat{\theta }_{q\tau }^{{T}},0^{{T}}\right] ^{{\mathrm{T}}}\), where \(\widehat{\beta }_{q\tau }\) and \(\widehat{\theta }_{q\tau }\) are defined as \(\widehat{\beta }_{q\tau },\widehat{\theta }_{q\tau }=\arg \min _{\beta _{\tau },\theta _{\tau }}\sum _{i=1}^{n}\frac{\delta _{i}}{\widehat{G}\left( \cdot \right) }\rho _{\tau }\left( Z_{i}-X_{1i}^{{T}}\beta _{\tau }-X_{3i} ^{{T}}\theta _{\tau }\right) ,\) that is \(\widehat{\beta }_{q\tau }\) and \(\widehat{\theta }_{q\tau }\) are the estimators of a parametric quantile regression, \(\left[ \beta _{\tau }^{0T},a_{\tau }^{0T},b_{\tau }^{0T}\right] ^{{\mathrm{T}}}=\left[ \widehat{\beta }_{q\tau }^{{T}},\widehat{a}_{f\tau }^{{T}},\widehat{b}_{f\tau }^{{T}}\right] ^{{\mathrm{T}}}\), where \(\widehat{a}_{f\tau }\) and \(\widehat{b}_{f\tau }\) are defined as \(\widehat{a}_{f\tau },\widehat{b}_{f\tau }=\arg \min _{a_{\tau },b_{\tau }}\sum _{i=1}^{n}\frac{\delta _{i}}{\widehat{G} \left( \cdot \right) }\rho _{\tau }\left( Z_{i}-X_{1i}^{{T}}\widehat{\beta }_{q\tau }-X_{3i}^{{T}}\left( a_{\tau f}-b_{\tau f}\left( X_{2i}-x_{2f}\right) \right) \right) K_{h}\left( X_{2i}-x_{2f}\right) \), where \(x_{2f}\) is a chosen point in the support of \(X_{2i}\) and the minimization is carried out using the Nelder–Mead algorithm, and finally \(\left[ \beta _{\tau } ^{0T},a_{\tau }^{0T},b_{\tau }^{0T}\right] ^{{\mathrm{T}}}\) are chosen as independent draws from a uniform distribution on \(\left( -\,2,2\right) .\) All of the above initial values resulted in final estimators with biases and/or IMSEs that were very close (with maximum difference at the second decimal place) to those reported in Tables 1, 2, 3, 4, 5, 6 and 7 in the paper.
I am grateful to one referee for suggesting this procedure.
To assess the sensitivity of the IMSE to this choice of b, we considered two alternative bandwidths, \(\widehat{b}_{1}=\widehat{b}/4\) and \(\widehat{b} _{2}=4\widehat{b},\) and computed the corresponding IMSE’s. The results of the simulations indicated that the IMSE’s of the resulting quantile estimators were still larger than those based on Breslow’s (1972) estimator.
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I am grateful to the Associate Editor and two Referees for useful comments and suggestions that improved considerably the paper. The usual disclaimer applies.
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Bravo, F. Semiparametric quantile regression with random censoring. Ann Inst Stat Math 72, 265–295 (2020). https://doi.org/10.1007/s10463-018-0688-3
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DOI: https://doi.org/10.1007/s10463-018-0688-3