Weighted censored quantile regression

Section 3. Estimation of weighted censored quantile regression parameters

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Define the distribution function of $T_i$ conditional on the $p$ -vector covariate, $X_i$ as $F_{T_i}\left(t|X_i\right)=\mathrm{Pr}\left(T_i\le t|X_i\right).$ Let ${\Lambda }_{T_i}\left(t|X_i\right)=-\log \left\{1-\mathrm{Pr}\left(T_i\le t|X_i\right)\right\},N_i\left(t\right)=\mathbb{I}\left(Y_i\le t,{\delta }_i=1\right),$ and $M_i\left(t\right)=N_i\left(t\right)-{\Lambda }_{T_i}\left(t\Lambda Y_i|X_i\right).$ Here ${\Lambda }_{T_i}\left(\cdot |X_i\right)$ is the cumulative hazard function conditional on $X_i,N_i\left(t\right)$ is the counting process and $M_i\left(t\right)$ is the martingale process associated with $N_i\left(t\right)$ (Fleming and Harrington, 2011). We consider an extension of censored quantile regression estimation procedure proposed by Peng and Huang (2008), incorporating the $P_i’s$ as known weights arrived based on the auxiliary information available through the known parameter $\theta .$ Note that $\theta$ and $\beta \left(\tau \right)$ are distinct parameters and estimating function $g\left(z:\theta \right)$ used for computing $P_i’s$ are different from the estimating functions used for quantile regression parameters in (1.1). Since $P_i’s$ are independent of $\beta \left(\tau \right),$ $E\left\{P_i{M}_i\left(t\right)|X_i\right\}=0$ (by the martingale property) for $t\ge 0,$ we have

$$E\left\{\sqrt{n}{\displaystyle \sum _{i=1}^n{P}_i{X}_i\left(N_i\left(e^{X_i^{\top }{\beta }_0\left(\tau \right)}\right)-{\Lambda }_T\left[e^{X_i^{\top }{\beta }_0\left(\tau \right)}\wedge Y_i|X_i\right]\right)}\right\}=0,(3.1)$$

where ${\beta }_0\left(\tau \right)$ denotes the true $\beta \left(\tau \right),$ in (1.2) for a given quantile, $\tau .$

Since ${\Lambda }_{T_i}\left(\cdot |X_i\right),i=1,2,\dots ,n$ are unknown functions, Peng and Huang (2008) derived the relationship between ${\Lambda }_T\left[e^{X_i^{\top }{\beta }_0\left(\tau \right)}\wedge Y_i|X_i\right]$ and ${\beta }_0\left(\tau \right)$ to use (3.1) to estimate ${\beta }_0\left(\tau \right).$ Using the fact that $F_{\mathfrak{F}}\left[e^{X_i^{\top }{\beta }_0\left(u\right)}|X_i\right]=\tau$ and utilizing the monotonicity of $X_i^T{\beta }_0\left(\tau \right)$ in $\tau ,$ they showed that ${\Lambda }_T\left[e^{X_i^{\top }{\beta }_0\left(\tau \right)}\wedge Y_i|X_i\right]={\displaystyle {\int }_0^{\tau }\mathbb{I}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right),}$ where $H\left(u\right)=-\log \left(1-u\right)$ for $0\le u<1.$

So, our weighted censored quantile regression estimating equation is

$$\sqrt{n}{S}_n\left(\beta ,\tau \right)=0,(3.2)$$

where

$$S_n\left(\beta ,\tau \right)={\displaystyle \sum _{i=1}^n{P}_i{X}_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)}\right\}.}$$

Here $P_i’s$ are defined in (2.2). Let $s\left(\beta ,\tau \right)=E\left\{S_n\left(\beta ,\tau \right)\right\}$ and the martingale property of $\mathbb{M}(\cdot )$ gives $s\left({\beta }_0,\tau \right)=0.$ For a particular quantile, ${\tau }_k$ and an estimator of ${\beta }_0\left({\tau }_k\right),\widehat{\beta }\left({\tau }_k\right)$ is a right-continuous step function which jumps only on a grid, ${\mathbb{S}}_L=\left\{0={\tau }_0<{\tau }_1<\dots <{\tau }_L={\tau }_U<1\right\}.$ Here $L$ depends on the sample size, $n.$ The size of ${\mathbb{S}}_L$ is defined as $\Vert {\mathbb{S}}_L\Vert =\underset{k}{\max }\left({\tau }_k-{\tau }_{k-1}\right).$

For different quantiles, ${\tau }_0,{\tau }_1,\dots ,{\tau }_L\left(0={\tau }_0<{\tau }_1<\dots <{\tau }_L<1\right),$ based on (3.2), we can obtain $\widehat{\beta }\left({\tau }_k\right)\left(k=1,2,\dots ,L\right)$ by recursively solving the following monotone estimating equation for $\beta \left({\tau }_k\right):$

$$\sqrt{n}{\displaystyle \sum _{i=1}^n{P}_i{X}_i}\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left({\tau }_k\right)}\right)-{\displaystyle \sum _{r=0}^{k-1}\mathbb{I}\left[Y_i\ge e^{X_i^{\top }\widehat{\beta }\left({\tau }_r\right)}\right]}\left\{H\left({\tau }_{r+1}\right)-H\left({\tau }_r\right)\right\}\right\}=0.(3.3)$$

We define the estimators, $\widehat{\beta }\left({\tau }_k\right)$ as the generalized solutions (Fygenson and Ritov, 1994) because equation (3.3) is not continuous and the solution may not be unique.

3.1  Asymptotic theory

We derived the asymptotic properties of the EL based weighted censored quantile regression estimators using the approach of Peng and Huang (2008). Now we prove the uniform consistency and weak Gaussian convergence of the proposed weighted censored quantile regression estimator, $\widehat{\beta }(\cdot ).$

Define $F\left(t|X\right)=\mathrm{Pr}\left(Y\le t|X\right),$ $\overline{F}\left(t|X\right)=\mathrm{Pr}\left(Y>t|X\right),$ $\tilde{F}\left(t|X\right)=\mathrm{Pr}\left(Y\le t,\delta =1|X\right),$ $\overline{f}\left(y|X\right)=-f\left(y|X\right)=-dF\left(y|X\right)/dy$ and $\tilde{f}\left(y|X\right)=d\tilde{F}\left(y|X\right)/dy.$ (For a vector $h,$ $h^{\otimes 2}=h{h}^T\text{},h^{\left(l\right)}=l^{\text{th}}$ component of $h,\Vert h\Vert$ is the Euclidean norm of $h.)$

Define $W_i={\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)X_i,$ $i=1,2,\dots ,n$ as a $p$ -vector.

Regularity conditions:

Theorem 1. Assuming that the regularity conditions R.1-R.7 hold, if $\underset{n\to \infty }{\lim }\Vert {\mathbb{S}}_L\Vert =0,$ then $\underset{\tau \in \left[v,{\tau }_U\right]}{\sup }\Vert \widehat{\beta }\left(\tau \right)-{\beta }_0\left(\tau \right)\Vert {\to }_{p}0,$ where $0<v<{\tau }_U.$

Theorem 2. Assuming that the regularity conditions R.1-R.7 hold, if $\underset{n\to \infty }{\lim }{n}^{1/2}\Vert {\mathbb{S}}_L\Vert =0,$ then $n^{1/2}\left\{\widehat{\beta }\left(\tau \right)-{\beta }_0\left(\tau \right)\right\}$ weakly converges to a zero-mean Gaussian process for $\tau \in \left[v,{\tau }_U\right],$ where $0<v<{\tau }_U.$

To prove Theorems 1 and 2, we need to show that $\underset{1\le i\le n}{\max }\left|{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right|=o_p\left(1\right).$ We consider two different types of $g\left(Z_i;\theta \right).$ First, $g\left(Z_i;\theta \right)$ does not contain the censored observations, as given in (4.1). The second, $g\left(Z_i;\theta \right),$ contains the censored observations, as given in (4.5).

In the case of uncensored observations, by Owen (1991) and Lemma 11.2 of Owen (2001), we have $\underset{1\le i\le n}{\max }\Vert g\left(Z_i;{\theta }_0\right)\Vert =o_p\left(\sqrt{n}\right).$ By Lemma 1 of Tang and Leng (2012), we have under the regularity conditions R.2, R.3; the ${\lambda }_{{\theta }_0}$ in (2.2) satisfies $\Vert {\lambda }_{{\theta }_0}\Vert =O_p\left(\frac{1}{\sqrt{n}}\right).$ So,

$$\underset{1\le i\le n}{\max }\left|{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right|=O_p\left(\frac{1}{\sqrt{n}}\right)o_p\left(\sqrt{n}\right)=o_p\left(1\right).(3.4)$$

Under the condition R.4; Qin and Jing (2001) proved $\underset{1\le i\le n}{\max }\left|{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right|=o_p\left(1\right)$ for the $g(\cdot )$ with censored observations.

Now following Owen (2001), using Taylor’s series expansion of the weights, $P_i’s$ defined in (2.2) can be rewritten as,

$$\begin{array}{ll}{P}_i\left({\theta }_0\right)\hfill & =\frac{1}{n\left\{1+{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right\}}\hfill \\ \hfill & =\frac{1}{n}\left[1-{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\left\{1+o_p\left(1\right)\right\}\right]\hfill \\ \hfill & =\frac{1}{n}\left[1-{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right]+o_p\left(\frac{1}{n}\right);i=1,2,\dots ,n.\hfill \end{array}$$

Now we rewrite the $S_n\left(\beta ,\tau \right)$ as

$$\begin{array}{ll}{S}_n\left(\beta ,\tau \right)\hfill & =\frac{1}{n}{\displaystyle \sum _{i=1}^n\left[1-{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)\right]X_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)\right\}+o_p\left(\frac{1}{n}\right)}\hfill \\ \hfill & =\frac{1}{n}{\displaystyle \sum _{i=1}^n{X}_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)\right\}}\hfill \\ \hfill & -\frac{1}{n}{\displaystyle \sum _{i=1}^n{\lambda }_{{\theta }_0}^{\top }g\left(Z_i;{\theta }_0\right)X_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)\right\}+o_p\left(\frac{1}{n}\right)}\hfill \\ \hfill & =\frac{1}{n}{\displaystyle \sum _{i=1}^n{X}_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)\right\}}\hfill \\ \hfill & -\frac{1}{n}{\displaystyle \sum _{i=1}^n{W}_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)\right\}+o_p\left(\frac{1}{n}\right).}\hfill \end{array}$$

Asymptotically, by (3.4) we have $\Vert W_i\Vert =o_p\left(1\right);i=1,2,\dots ,n.$ So,

$$S_n\left(\beta ,\tau \right)=\frac{1}{n}{\displaystyle \sum _{i=1}^n{X}_i\left\{{\mathbb{N}}_i\left(e^{X_i^{\top }\beta \left(\tau \right)}\right)-{\displaystyle {\int }_0^{\tau }\mathbb{I}}\left[Y_i\ge e^{X_i^{\top }\beta \left(u\right)}\right]dH\left(u\right)\right\}+o_p\left(\frac{1}{n}\right).}$$

Asymptotically this estimating function, $S_n\left(\beta ,\tau \right)$ is equivalent to that in Peng and Huang (2008). Following the similar arguments of Peng and Huang (2008), we complete the proofs of Theorems 1 and 2.

As indicated in Peng and Huang (2008), the estimation of asymptotic variance of the quantile regression estimates is not easy since the covariance matrix of the limiting process of $\sqrt{n}\left\{\widehat{\beta }\left(\tau \right)-{\beta }_0\left(\tau \right)\right\}$ involves unknown density function $f\left(y|X\right)$ and $\tilde{f}\left(y|X\right).$ Instead of using a smoothing or other numerical approximations, we suggest a simple bootstrap approach to estimate the standard errors of the regression estimates. This approach is used in our performance analysis discussed in next section.

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