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.