These are objects of class rq.process.
They represent the fit of a linear conditional quantile function model.
This class of objects is returned from the rq
function
to represent a fitted linear quantile regression model.
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
The following components must be included in a legitimate rq.process
object.
solThe 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 \(J=O_p(n \log n)\).
dsolThe 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).
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.
[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.
rq.