The HPQUANTSELECT Procedure

Diagnostic Statistics

This section gathers the formulas for the statistics available in the OUTPUT statement. All the statistics available in the OUTPUT statement are conditional on the selected model and do not take into account the variability that is introduced by doing model selection.

The model to be fit is upper Q Subscript upper Y Baseline left-parenthesis tau vertical-bar bold x right-parenthesis equals bold x prime bold-italic beta left-parenthesis tau right-parenthesis, and the parameter estimate ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis is the solution that minimizes sigma-summation Underscript i equals 1 Overscript n Endscripts rho Subscript tau Baseline left-parenthesis y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta right-parenthesis. The subscript i is for the ith observation. The subscript j is for the jth-smallest quantile level among all the specified QUANTILE= levels in the MODEL statement. ModifyingAbove normal upper Sigma With caret left-parenthesis tau right-parenthesis denotes the covariance estimation for ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis.

The ALPHA= option in the PROC HPQUANTSELECT statement sets the alpha value for the confidence limit statistics. The degrees of freedom for t Subscript StartFraction alpha Over 2 EndFraction are n minus p.

Table 7 contains the diagnostic statistics and their formulas. Each statistic is computed for each observation.

Table 7: Formulas and Definitions for Diagnostic Statistics

MODEL Option or Statistic Formula
PREDj ModifyingAbove y With caret Subscript j i Baseline equals bold x prime Subscript i Baseline ModifyingAbove bold-italic beta With caret left-parenthesis tau Subscript j Baseline right-parenthesis
RESj y Subscript i Baseline minus ModifyingAbove y With caret Subscript j i
STDPj StartRoot bold x prime Subscript i Baseline ModifyingAbove normal upper Sigma With caret left-parenthesis tau Subscript j Baseline right-parenthesis bold x Subscript i Baseline EndRoot
LCLMj ModifyingAbove y With caret Subscript j i Baseline minus t Subscript StartFraction alpha Over 2 EndFraction Baseline STDP j Subscript i Baseline
UCLMj ModifyingAbove y With caret Subscript j i Baseline plus t Subscript StartFraction alpha Over 2 EndFraction Baseline STDP j Subscript i Baseline


Last updated: December 09, 2022