The HPQUANTSELECT Procedure

More Statistics for Parameter Estimates

Let COV left-parenthesis dot right-parenthesis denote the covariance function matrix of a random vector. Then, the sparsity-function estimates of COV left-parenthesis ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis right-parenthesis is

ModifyingAbove COV With caret left-parenthesis ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column omega squared left-parenthesis tau comma upper F right-parenthesis bold upper Omega Superscript negative 1 Baseline slash n 2nd Column for a linear model with iid errors 2nd Row 1st Column tau left-parenthesis 1 minus tau right-parenthesis bold upper H Subscript n Superscript minus Baseline bold upper Omega bold upper H Subscript n Superscript minus Baseline slash n 2nd Column for a linear hyphen in hyphen parameter model with non hyphen iid settings EndLayout

where ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis equals left-parenthesis ModifyingAbove beta With caret Subscript 1 Baseline left-parenthesis tau right-parenthesis comma ellipsis comma ModifyingAbove beta With caret Subscript p Baseline left-parenthesis tau right-parenthesis right-parenthesis is the vector of the parameter estimates.

If you specify the CLB option in the MODEL statement, PROC HPQUANTSELECT outputs the standard error, confidence limits, t value, and Pr > StartAbsoluteValue t EndAbsoluteValue probability for each ModifyingAbove beta With caret Subscript j Baseline left-parenthesis tau right-parenthesis in the parameter estimates table. Table 5 summarizes these statistics for ModifyingAbove beta With caret Subscript j Baseline left-parenthesis tau right-parenthesis.

Table 5: More Statistics for ModifyingAbove beta With caret Subscript j Baseline left-parenthesis tau right-parenthesis

Statistic Definition
Standard error: ModifyingAbove sigma With caret Subscript j StartRoot ModifyingAbove COV With caret left-parenthesis ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis right-parenthesis Subscript j j Baseline EndRoot
left-parenthesis 1 minus alpha right-parenthesis% confidence limits left-parenthesis ModifyingAbove beta With caret Subscript j Baseline left-parenthesis tau right-parenthesis plus-or-minus t Subscript 1 comma 1 minus StartFraction alpha Over 2 EndFraction Baseline ModifyingAbove sigma With caret Subscript j Baseline right-parenthesis
t value ModifyingAbove beta With caret Subscript j Baseline left-parenthesis tau right-parenthesis slash ModifyingAbove sigma With caret Subscript j
Pr > StartAbsoluteValue t EndAbsoluteValue probability p-value of the t value


Here COV left-parenthesis ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis right-parenthesis Subscript j j Baseline is the left-parenthesis j comma j right-parenthesis element of COV left-parenthesis ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis right-parenthesis, and t Subscript 1 comma 1 minus StartFraction alpha Over 2 EndFraction denotes the left-parenthesis 1 minus StartFraction alpha Over 2 EndFraction right-parenthesis-level student’s t score with 1 degree of freedom.

Last updated: December 09, 2022