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
.