Linear Quantile Regression Process Object

Description

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

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.

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, …, 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).

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.