Weighted censored quantile regression

Section 2. Estimation of weights using empirical likelihood

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We develop a method that converts the auxiliary information to the EL based data driven probabilities, which are further used in the weighted censored quantile regression as the weights. Qin and Lawless (1994) developed the EL approach based on a set of estimating equations. Let ${\left\{Z_i\right\}}_{i=1}^n$ be the observed data and the available auxiliary information is represented by the estimating function $g\left(Z_i;\theta \right)$ with parameter, $\theta$ which is known. Then, the maximum empirical likelihood is given by

$$L_{\text{EL}}\left(\theta \right)=\sup \left\{{\displaystyle \prod _{i=1}^n{P}_i:P_i\ge 0,}{\displaystyle \sum _{i=1}^n{P}_i=1,}{\displaystyle \sum _{i=1}^n{P}_{i}g\left(Z_i;\theta \right)=0}\right\},(2.1)$$

where $P_i=\mathrm{Pr}\left(Z_i=z_i\right)$ and $\theta$ is the parameter in the auxiliary information which can be assumed to be known. The parameter, $\theta$ could be any parametric information available from the previous studies which has an influence on the model parameter, $\beta \left(\tau \right).$ For a given $g\left(Z_i;\theta \right),\theta$ should satisfy $E\left\{g\left(Z_i;\theta \right)\right\}=0$ to avoid the non-existence of solutions due to convex hull issues. This is the scenario for when zero is not an inner point of the convex hull of the $g\left(Z_i;\theta \right),i=1,2,\dots ,n,$ which will fail to provide positive $P_i’s.$ For a given value of $\theta ={\theta }_0,$ using the Lagrange multiplier method, $L_{\text{EL}}\left({\theta }_0\right)$ attains its maximum at

$$P_i\left({\theta }_0\right)=\frac{1}{n\left\{1+{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right\}},i=1,2,\dots ,n.(2.2)$$

The Lagrange multiplier, ${\lambda }_{{\theta }_0}$ is the solution to the equation

$${\displaystyle \sum _{i=1}^n\frac{g\left(Z_i;{\theta }_0\right)}{n\left\{1+{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right\}}}=0.$$

The estimated $P_i(\cdot )’s$ are used as the weights $\left({\omega }_i\right)$ in (1.2) for the weighted censored quantile regression. In some cases, $\theta$ may not be available and in such situations, we can use an estimate of $\theta ,$ say ${\widehat{\theta }}_A$ obtained from previous studies. Chen and Qin (1993) showed that for a random sample, $Y_i,$ and $P_i(\cdot )’s$ are estimated using (2.2), ${\tilde{F}}_n\left(y\right)={\displaystyle {\sum }_{i=1}^n{P}_i\mathbb{I}\left(Y_i\le y\right)}$ has smaller variance than the empirical distribution function, ${\widehat{F}}_n\left(y\right)=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n\mathbb{I}\left(Y_i\le y\right)}.$ As a result, with Bahadur representation (Bahadur, 1966), for a given $\tau \left(0<\tau <1\right),$ the quantile estimate, ${\tilde{F}}_n^{-1}\left(\tau \right)$ has smaller variance than ${\widehat{F}}_n^{-1}\left(\tau \right)$ (See Kuk and Mak, 1989; Rao et al., 1990). Hence our proposed method is expected to be more efficient than the ordinary censored quantile regression.

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