Quantile function regression analysis for interval censored data, with application to salary survey data

Abstract

This study aims at regression analysis for quantile functions where the quantile regression coefficients are treated as functions over a continuum of quantile levels. We propose a general inference procedure for quantile regression coefficient functions with interval-censored outcome data. The modeling framework follows a recent proposal using a set of parametric basis functions to approximate the quantile regression coefficient functions. The new proposal can accommodate outcome data subject to general types of interval censoring, including fixed, random, and partly interval censoring. The large sample theory for the proposed estimator is established for inference, and a goodness-of-fit testing procedure is developed to guide the choice of the basis functions. We apply the proposed methodology to a survey dataset on monthly salaries of Taiwan workers, where only parts of the salary data are exact while the others are interval-censored according to the salary intervals prespecified in the survey questionnaire.

Appendix

1.1 Derivation of Equation (2.2)

Since $\tau _{y_i}=\{\tau :\,Q(\tau ;{\varvec{x}}_i)={\varvec{x}}_i^{\mathrm{T}}{\varvec{\beta }}(\tau )=y_i\}$ is assumed to be unique and $Q(\tau ;{\varvec{x}}_i)$ is a monotone function of $\tau$, we have $I(y_i\le {\varvec{x}}_i^{\mathrm{T}}{\varvec{\beta }}(\tau )<\infty )=I(\tau _{y_i}\le \tau \le 1)$. Then,

$$\begin{aligned} {\varvec{0}}=&\sum _{i=1}^n \int _0^1 ({\varvec{b}}(\tau )\otimes {\varvec{x}}_i){S}_i(\tau ){\text {d}}\tau \\ =&\sum _{i=1}^n \Big \{\int _0^1 {\varvec{b}}(\tau )I(\tau _{y_i}\le \tau \le 1){\text {d}}\tau -\int _0^1 \tau {\varvec{b}}(\tau ) {\text {d}}\tau \Big \} \otimes {\varvec{x}}_i\\ =&\sum _{i=1}^n \left\{ {{\varvec{B}}}(1)-{{\varvec{B}}}(\tau _{y_i})-\bar{{\varvec{B}}}(1)\right\} \otimes {\varvec{x}}_i. \end{aligned}$$

1.2 Proof of $E\{{{\tilde{e}}}_i(\tau )\}=\tau$

It is sufficient to prove that $E\{e_i(\tau )\}=\tau$. For ease of presentation, we remove the subscript i of $c_{i1},c_{i2}$ and ${\varvec{x}}_i$, and write $F_Y$ short for $F_{Y|{\varvec{x}}}$.

$$\begin{aligned} E\{I(c_2 \le {\varvec{x}}^{\mathrm{T}}{\varvec{\beta }})\}&=\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{-\infty }^{c_2} f_{c_1,c_2}(c_1,c_2){\text {d}}c_1{\text {d}}c_2\nonumber \\&=\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\Big \{\int _{-\infty }^{c_2}\frac{F_Y(c_2)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_1\Big \}{\text {d}}c_2\nonumber \\&\quad -\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\Big \{\Big (\int _{c_1}^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}} +\int _{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}^{\infty }\Big ) \frac{F_Y(c_1)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2\Big \}{\text {d}}c_1\nonumber \\&\quad +\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\Big \{\int _{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}^{\infty } \frac{F_Y(c_1)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2\Big \}{\text {d}}c_1\nonumber \\&= - \int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}F_Y(c) d\Big \{\int _{-\infty }^{c}\int _{c}^{\infty }\frac{f_{c_1,c_2}(c_1,c_2)}{F_Y(c_2)-F_Y(c_1)}{\text {d}}c_2{\text {d}}c_1\Big \}\nonumber \\&\quad +\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}^{\infty } \frac{F_Y(c_1)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2{\text {d}}c_1. \end{aligned}$$

(7.1)

By integration by parts,

$$\begin{aligned}&-\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}F_Y(c) d\Big \{\int _{-\infty }^{c}\int _{c}^{\infty }\frac{f_{c_1,c_2}(c_1,c_2)}{F_Y(c_2)-F_Y(c_1)}{\text {d}}c_2{\text {d}}c_1\Big \}\nonumber \\&\quad =\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}^{\infty } \frac{-\tau }{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2{\text {d}}c_1 \nonumber \\&\qquad +\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{-\infty }^c\int _c^{\infty } \frac{f_Y(c)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2{\text {d}}c_1{\text {d}}c. \end{aligned}$$

(7.2)

and

$$\begin{aligned}&\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{-\infty }^c\int _c^{\infty } \frac{f_Y(c)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2{\text {d}}c_1{\text {d}}c\nonumber \\&\quad =\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{-\infty }^c\int _c^{\infty } f_{Y|\,c_1,c_2}(c|\,c_1,c_2)f_{c_1,c_2}(c_1,c_2){\text {d}}c_2{\text {d}}c_1{\text {d}}c=P(Y \le {\varvec{x}}^{\mathrm{T}}{\varvec{\beta }})=\tau . \end{aligned}$$

(7.3)

It follows from (7.1) – (7.3), together with

$$\begin{aligned} E\Big \{\frac{\tau -F_Y(c_1)}{F_Y(c_2)-F_Y(c_1)}I(c_1< {\varvec{x}}^{\mathrm{T}}{\varvec{\beta }} < c_2)\Big \} =\int _{-\infty }^{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}\int _{{\varvec{x}}^{\mathrm{T}}{\varvec{\beta }}}^{\infty } \frac{\tau -F_Y(c_1)}{F_Y(c_2)-F_Y(c_1)}f_{c_1,c_2}(c_1,c_2){\text {d}}c_2{\text {d}}c_1 \end{aligned}$$

that we prove $E\{e_i(\tau )\}=\tau$.