# The QUANTLIFE Procedure

### Kaplan-Meier-Type Estimator for Censored Quantile Regression

Portnoy (2003) proposes the use of weighted quantile regression to sequentially estimate along the equally spaced grid . You can request this method by specifying the METHOD=KM option in the PROC QUANTLIFE statement. The grid points are equally spaced, with specified by the INITTAU= option and the step between adjacent grid points specified by the GRIDSIZE=option.

This method uses a weight function for each censored observation. The weight function is constructed as follows: Let be the first grid point at which and ; otherwise let . When computing the th quantile, assign weight to the censored observation if ; otherwise assign . The algorithm for computing is as follows:

1. Compute by using the standard quantile regression method.

2. For , obtain sequentially by minimizing the following weighted quantile regression objective function:

where is the weight for the right-censored observation at computing , and the complementary weight is for , a large constant that is greater than all .

The weighted quantile regression method is similar to Efron’s redistribution-of-mass idea (Efron 1967) for the Kaplan-Meier estimator.

Note that if all observations are uncensored, is the same as the standard quantile regression estimator.