<!doctype s-function-doc system "s-function-doc.dtd" [
<!entity % S-OLD "INCLUDE">
]
>
<s-function-doc>
<s-topics>
   <s-topic>rq.object</s-topic>
</s-topics>
<s-title>
Linear Quantile Regression Process Object 
</s-title>
<s-description>
These are objects of class
<s-expression>
rq.process. 
</s-expression>
They represent the fit of a linear conditional quantile function model. 
</s-description>
<s-section name=" GENERATION">
This class of objects is returned from the 
<s-expression>rq</s-expression> function 
to represent a fitted linear quantile regression model. 
</s-section>
<s-section name=" METHODS">
The <s-expression>"rq.process"</s-expression> class of objects has 
methods for the following generic 
functions: 
<br><br>
<s-expression>effects</s-expression>, 
<s-expression>formula</s-expression>, 
<s-expression>labels</s-expression>,  
<s-expression>model.frame</s-expression>, 
<s-expression>model.matrix</s-expression>, 
<s-expression>plot</s-expression>, 
<s-expression>predict</s-expression>, 
<s-expression>print</s-expression>, 
<s-expression>print.summary</s-expression>, 
<s-expression>summary</s-expression>,  
<br><br>
</s-section>
<s-section name=" STRUCTURE">
The following components must be included in a legitimate 
<s-expression>rq.process</s-expression> object. 
</s-section>
<s-value>
<s-return-component name="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).
</s-return-component>
<s-return-component name="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).
</s-return-component>
</s-value>
<s-details>
These arrays are computed by parametric linear programming methods
using the exterior point (simplex-type) methods of the 
Koenker-d'Orey algorithm based on Barrodale and Roberts median
regression algorithm.
</s-details>
<s-section name=" REFERENCES">


[1] Koenker, R.W. and Bassett, G.W. (1978). Regression quantiles,
<it>Econometrica</it>, 46, 33-50.
<br><br>
[2] Koenker, R.W. and d'Orey (1987,1994). Computing Regression Quantiles.
<it>Applied Statistics</it>, 36, 383-393, and 43, 410-414.
<br><br>
[3] Gutenbrunner, C. Jureckova, J. (1991).
Regression quantile and regression rank score process in the
linear model and derived statistics, <it>Annals of Statistics</it>, 20, 305-330.
<br><br>
[4] Gutenbrunner,  C.,  J.  Jureckova,   Koenker,  R.  and
Portnoy,  S.(1994)  "Tests  of Linear Hypotheses  based on Regression  
Rank   Scores",   <it>Journal   of   Nonparametric Statistics</it>, 
(2), 307-331.
<br><br>
[5]  Portnoy, S. (1991).  Asymptotic behavior of the number of regression
quantile breakpoints, <it> SIAM Journal of  Scientific
and  Statistical Computing</it>, 12, 867-883.
</s-section>
<s-see>
<s-function name="rq.sgm">rq</s-function>.</s-see>
<s-keywords>
<s-keyword>librobust</s-keyword>
<s-keyword>classes</s-keyword>
</s-keywords>
<s-docclass>
function
</s-docclass>
</s-function-doc>
