The QUANTSELECT Procedure

Observation Quantile Level

The observation quantile level of a valid observation, left-parenthesis y comma bold x right-parenthesis, is defined as tau Subscript left-parenthesis y comma bold x right-parenthesis Baseline equals upper F Subscript upper Y vertical-bar bold x Baseline left-parenthesis y right-parenthesis, where upper F Subscript upper Y vertical-bar bold x Baseline left-parenthesis dot right-parenthesis denotes the cumulative distribution function (CDF) for the y’s underlying distribution conditional on bold x. For the CDF that is continuous at y, the equation y equals upper Q Subscript upper Y vertical-bar bold x Baseline left-parenthesis tau Subscript left-parenthesis y comma bold x right-parenthesis Baseline right-parenthesis holds because the quantile function is inversely related to the CDF. Ideally, if y equals bold x ModifyingAbove bold-italic beta With caret left-parenthesis tau Superscript asterisk Baseline right-parenthesis for a unique tau Superscript asterisk Baseline element-of left-bracket 0 comma 1 right-bracket and some quantile-regression optimal solution ModifyingAbove bold-italic beta With caret left-parenthesis tau Superscript asterisk Baseline right-parenthesis, then tau Superscript asterisk is a reasonable estimation for tau Subscript left-parenthesis y comma bold x right-parenthesis, written as ModifyingAbove tau With caret Subscript left-parenthesis y comma bold x right-parenthesis Baseline equals tau Superscript asterisk. However, such a tau Superscript asterisk might not exist or is nonunique in practice. The following steps show how the QUANTSELECT procedure estimates the observation quantile level tau Subscript left-parenthesis y comma bold x right-parenthesis via quantile process regression:

  1. Fit the quantile process regression model and label its quantile-level grid as follows:

    StartSet 0 equals tau Subscript left-parenthesis 0 right-parenthesis Baseline less-than-or-equal-to tau Subscript left-parenthesis 1 right-parenthesis Baseline less-than-or-equal-to midline-horizontal-ellipsis less-than-or-equal-to tau Subscript left-parenthesis s right-parenthesis Baseline less-than-or-equal-to tau Subscript left-parenthesis s plus 1 right-parenthesis Baseline equals 1 EndSet

  2. Compute quantile predictions conditional on bold x in the quantile-level grid: StartSet q Subscript i Baseline equals bold x ModifyingAbove bold-italic beta With caret Subscript i Baseline colon i equals 0 comma ellipsis comma s plus 1 EndSet.

  3. Sort q Subscript i’s to avoid crossing, such that q Subscript left-parenthesis 0 right-parenthesis Baseline less-than-or-equal-to q Subscript left-parenthesis 1 right-parenthesis Baseline less-than-or-equal-to midline-horizontal-ellipsis less-than-or-equal-to q Subscript left-parenthesis s plus 1 right-parenthesis.

  4. ModifyingAbove tau With caret Subscript left-parenthesis y comma bold x right-parenthesis Baseline equals 0 if y less-than q Subscript left-parenthesis 0 right-parenthesis, or ModifyingAbove tau With caret Subscript left-parenthesis y comma bold x right-parenthesis Baseline equals 1 if y greater-than q Subscript left-parenthesis s plus 1 right-parenthesis.

  5. Otherwise, search index j such that q Subscript left-parenthesis j right-parenthesis Baseline less-than y less-than q Subscript left-parenthesis j plus 1 right-parenthesis. If such a j exists,

    ModifyingAbove tau With caret Subscript left-parenthesis y comma bold x right-parenthesis Baseline equals left-parenthesis StartFraction y minus q Subscript left-parenthesis j right-parenthesis Baseline Over q Subscript left-parenthesis j plus 1 right-parenthesis Baseline minus q Subscript left-parenthesis j right-parenthesis Baseline EndFraction right-parenthesis tau Subscript left-parenthesis j plus 1 right-parenthesis Baseline plus left-parenthesis StartFraction q Subscript left-parenthesis j plus 1 right-parenthesis Baseline minus y Over q Subscript left-parenthesis j plus 1 right-parenthesis Baseline minus q Subscript left-parenthesis j right-parenthesis Baseline EndFraction right-parenthesis tau Subscript left-parenthesis j right-parenthesis

  6. Otherwise, search j and k such that q Subscript left-parenthesis j minus 1 right-parenthesis Baseline less-than y equals q Subscript left-parenthesis j right-parenthesis Baseline equals midline-horizontal-ellipsis equals q Subscript left-parenthesis j plus k right-parenthesis Baseline less-than q Subscript left-parenthesis j plus k plus 1 right-parenthesis, and set ModifyingAbove tau With caret Subscript left-parenthesis y comma bold x right-parenthesis Baseline equals StartFraction tau Subscript left-parenthesis j right-parenthesis Baseline plus tau Subscript left-parenthesis j plus k right-parenthesis Baseline Over 2 EndFraction. Here, define q Subscript left-parenthesis negative 1 right-parenthesis Baseline equals negative normal infinity and q Subscript left-parenthesis s plus 2 right-parenthesis Baseline equals normal infinity.

Last updated: December 09, 2022