The observation quantile level of a valid observation, , is defined as , where denotes the cumulative distribution function (CDF) for the y’s underlying distribution conditional on . For the CDF that is continuous at y, the equation holds because the quantile function is inversely related to the CDF. Ideally, if for a unique and some quantile-regression optimal solution , then is a reasonable estimation for , written as . However, such a might not exist or is nonunique in practice. The following steps show how the QUANTSELECT procedure estimates the observation quantile level via quantile process regression:
Fit the quantile process regression model and label its quantile-level grid as follows:
Compute quantile predictions conditional on in the quantile-level grid: .
Sort ’s to avoid crossing, such that .
if , or if .
Otherwise, search index j such that . If such a j exists,
Otherwise, search j and k such that , and set . Here, define and .