The QUANTLIFE Procedure

Nelson-Aalen-Type Estimator for Censored Quantile Regression

Peng and Huang (2008) propose a method of censored quantile regression that is based on the Nelson-Aalen estimator of the cumulative hazard function. Let upper F Subscript i Baseline left-parenthesis t vertical-bar x right-parenthesis equals upper P left-parenthesis upper T Subscript i Baseline less-than-or-equal-to t vertical-bar x Subscript i Baseline right-parenthesis comma normal upper Lamda Subscript i Baseline left-parenthesis t vertical-bar x right-parenthesis equals minus normal l normal o normal g left-parenthesis 1 minus upper F Subscript i Baseline left-parenthesis t vertical-bar x right-parenthesis right-parenthesis, and upper N Subscript i Baseline left-parenthesis t right-parenthesis equals upper I left-brace StartSet upper T Subscript i Baseline less-than-or-equal-to t EndSet and StartSet normal upper Delta Subscript i Baseline equals 1 EndSet right-brace. Then the following equation is a martingale process that is associated with the counting process upper N Subscript i Baseline left-parenthesis t right-parenthesis (Fleming and Harrington 1991):

upper M Subscript i Baseline left-parenthesis t right-parenthesis equals upper N Subscript i Baseline left-parenthesis t right-parenthesis minus normal upper Lamda Subscript i Baseline left-parenthesis t logical-and upper Y Subscript i Baseline vertical-bar x right-parenthesis

Based on the martingale process, Peng and Huang (2008) derive the following estimating equation:

n Superscript negative 1 slash 2 Baseline sigma-summation Underscript i equals 1 Overscript n Endscripts x Subscript i Baseline left-bracket upper N Subscript i Baseline left-parenthesis normal e normal x normal p left-parenthesis x prime Subscript i Baseline beta left-parenthesis tau right-parenthesis right-parenthesis right-parenthesis minus integral Subscript 0 Superscript tau Baseline upper I left-parenthesis upper Y Subscript i Baseline greater-than-or-equal-to normal e normal x normal p left-parenthesis x prime Subscript i Baseline beta left-parenthesis tau right-parenthesis right-parenthesis right-parenthesis d upper H left-parenthesis u right-parenthesis right-bracket equals 0

where upper H left-parenthesis u right-parenthesis equals minus normal l normal o normal g left-parenthesis 1 minus u right-parenthesis and u element-of left-bracket 0 comma 1 right-parenthesis. By approximating the integral in the estimating equation on a grid 0 equals tau 0 less-than tau 1 less-than midline-horizontal-ellipsis less-than tau Subscript upper M Baseline less-than 1, the regression quantiles beta left-parenthesis tau Subscript k Baseline right-parenthesis, k equals 1 comma ellipsis comma upper M, can be estimated sequentially by solving the following linear programming problem:

min Underscript b Endscripts left-brace alpha left-parenthesis tau Subscript k Baseline right-parenthesis prime u plus left-parenthesis normal upper Delta minus alpha left-parenthesis tau Subscript k Baseline right-parenthesis right-parenthesis prime v vertical-bar z equals upper X b plus u minus v comma u greater-than-or-equal-to 0 comma v greater-than-or-equal-to 0 right-brace

where

alpha left-parenthesis tau Subscript k Baseline right-parenthesis equals sigma-summation Underscript j equals 1 Overscript k minus 1 Endscripts upper I left-parenthesis upper Y Subscript i Baseline greater-than-or-equal-to normal e normal x normal p left-parenthesis x prime Subscript i Baseline ModifyingAbove beta With caret left-parenthesis tau Subscript j Baseline right-parenthesis right-parenthesis right-parenthesis upper H left-parenthesis left-parenthesis u Subscript j plus 1 Baseline right-parenthesis minus upper H left-parenthesis u Subscript j Baseline right-parenthesis right-parenthesis

and X is the known matrix of x Subscript i’s. For more information, see Koenker (2008).

You can request this method by specifying the METHOD=NA option. The grid points 0 equals tau 0 less-than tau 1 less-than midline-horizontal-ellipsis less-than tau Subscript upper M Baseline less-than 1 are equally spaced, with tau 1 specified by the INITTAU=option and the grid step between two adjacent grid points specified by the GRIDSIZE=option.

Last updated: December 09, 2022