quantreg (version 5.54)

rq.process.object: Linear Quantile Regression Process Object

Description

These are objects of class rq.process. They represent the fit of a linear conditional quantile function model.

Arguments

Generation

This class of objects is returned from the rq function to represent a fitted linear quantile regression model.

Methods

The "rq.process" class of objects has methods for the following generic functions: effects, formula , labels , model.frame , model.matrix , plot , predict , print , print.summary , summary

Structure

The following components must be included in a legitimate rq.process object.

sol

The primal solution array. This is a (p+3) by J matrix whose first row contains the 'breakpoints' \(tau_1, tau_2, \dots, tau_J\), of the quantile function, i.e. the values in [0,1] at which the solution changes, row two contains the corresponding quantiles evaluated at the mean design point, i.e. the inner product of xbar and \(b(tau_i)\), the third row contains the value of the objective function evaluated at the corresponding \(tau_j\), and the last p rows of the matrix give \(b(tau_i)\). The solution \(b(tau_i)\) prevails from \(tau_i\) to \(tau_i+1\). Portnoy (1991) shows that \(J=O_p(n \log n)\).

dsol

The dual solution array. This is a n by J matrix containing the dual solution corresponding to sol, the ij-th entry is 1 if \(y_i > x_i b(tau_j)\), is 0 if \(y_i < x_i b(tau_j)\), and is between 0 and 1 otherwise, i.e. if the residual is zero. See Gutenbrunner and Jureckova(1991) for a detailed discussion of the statistical interpretation of dsol. The use of dsol in inference is described in Gutenbrunner, Jureckova, Koenker, and Portnoy (1994).

Details

These arrays are computed by parametric linear programming methods using using the exterior point (simplex-type) methods of the Koenker--d'Orey algorithm based on Barrodale and Roberts median regression algorithm.

References

[1] Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles, Econometrica, 46, 33--50.

[2] Koenker, R. W. and d'Orey (1987, 1994). Computing Regression Quantiles. Applied Statistics, 36, 383--393, and 43, 410--414.

[3] Gutenbrunner, C. Jureckova, J. (1991). Regression quantile and regression rank score process in the linear model and derived statistics, Annals of Statistics, 20, 305--330.

[4] Gutenbrunner, C., Jureckova, J., Koenker, R. and Portnoy, S. (1994) Tests of linear hypotheses based on regression rank scores. Journal of Nonparametric Statistics, (2), 307--331.

[5] Portnoy, S. (1991). Asymptotic behavior of the number of regression quantile breakpoints, SIAM Journal of Scientific and Statistical Computing, 12, 867--883.

See Also

rq.