.BG
.FN rq
.TL
Quantile Regression
.DN
Perform a quantile regression on a design matrix, x, of explanatory variables and a vector, y, of responses. 
.CS
rq(x, y, tau=-1, alpha=.1, dual=F, int=T, ci = T, method="score",
interpolate=T, tcrit=T, hs=T)
.PP
.RA
.AG x
vector or matrix of explanatory variables.  If  a  matrix,
each  column represents a variable and each row represents
an observation (or case).  This should not contain  column
of  1s unless the argument intercept is FALSE.  The number
of rows of x should equal the number of elements of  y,  and
there  should  be fewer columns than rows.  
If x is missing, rq() computes the ordinary
sample quantile(s) of y.
.AG y
response vector with as many observations as the number of rows of x. 
.OA
.AG tau
desired quantile. If tau is missing or outside the range [0,1]
then all the regression quantiles are computed and the corresponding primal and dual solutions are returned.
.AG alpha
level of significance for the confidence intervals; default is set at 10%.
.AG dual
return the dual solution if TRUE (default).
.AG int
flag for intercept; if TRUE (default) an intercept term is included in the regression.
.AG ci
flag for confidence interval; if TRUE (default) the confidence intervals are 
returned.  If tau is outside [0,1], ci is automatically set to FALSE.
.AG method
if method="score" (default), ci is computed using regression rank score inversion;
if method="sparsity", ci is computed using sparsity function.
.AG interpolate
if TRUE (default), the smoothed confidence intervals are returned.
.AG tcrit
if tcrit=T (default), a finite sample adjustment of the critical point is
performed using Student's t quantile, else the standard Gaussian quantile is 
used.
.AG hs
logical flag to use Hall-Sheather's sparsity estimator (default); otherwise Bofinger's
version is used.
.RT If tau is in [0,1] the function returns:
.RC coef
the estimated parameters of the tau-th conditional quantile function.
.RC resid
the estimated residuals of the tau-th conditional quantile function.
.RC dual
the dual solution (if dual=T).
.RC h
the index of observations in the basis.
.RC ci
confidence intervals (if ci=T).
.RT If tau isn't in [0,1] the function returns:
.RC sol
a  (p+2) by m matrix whose first row contains the 'breakpoints'
tau_1,tau_2,...tau_m, 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), 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.
.RC dsol
the matrix of dual solutions corresponding to the primal solutions in sol.
This is an n by m matrix whose 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.
.RC h
the matrix of observations indices in the basis corresponding to sol or dsol.
.EX
rq(stack.x,stack.loss,.5)  #the l1 estimate for the stackloss data
rq(stack.x,stack.loss,tau=.5,ci=T,method="score")  #same as above with 
	#regression rank score inversion confidence interval
rq(stack.x,stack.loss,.25)  #the 1st quartile, 
	#note that 8 of the 21 points lie exactly 
	#on this plane in 4-space
rq(stack.x,stack.loss,-1)   #this gives all of the rq solutions
rq(y=rnorm(10),method="sparsity")	#ordinary sample quantiles
.SH METHOD
The algorithm used is a modification of the Barrodale and Roberts
algorithm for l1-regression, l1fit in S, and is described in detail
in Koenker and d"Orey(1987).
.KW regression quantiles, robust estimation, L-estimators
.SH REFERENCES
[1] Koenker, R.W. and Bassett, G.W. (1978). Regression quantiles, Econometrica, 46, 33-50.

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

[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] Koenker, R.W. and d'Orey (1994).  Remark on Alg. AS 229: Computing Dual
Regression Quantiles and Regression Rank Scores, Applied Statistics, 43, 410-414.

[5] Koenker, R.W. (1994). Confidence Intervals for Regression Quantiles, in
P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349-359, Springer-Verlag,
New York.

.SH SEE ALSO
trq and qrq for further details and references.
.PP
.WR