

   QQuuaannttiillee RReeggrreessssiioonn

        rq(x, y, tau=-1, alpha=.1, dual=TRUE, int=TRUE, tol=1e-4, ci = TRUE,
                 method="score", interpolate=TRUE, tcrit=TRUE, hs=TRUE)
        rq.formula(formula, data=list(), subset, na.action, tau=-1,
                 alpha = 0.10000000000000001, dual = TRUE,
                 tol = 0.0001, ci = TRUE, method="score", interpolate = TRUE,
                 tcrit = TRUE, hs=TRUE)

   AArrgguummeennttss::

          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.

          y: response vector with as many observations as the
             number of rows of x.

        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.

      alpha: level of significance for the confidence inter-
             vals; default is set at 10%.

       dual: return the dual solution if TRUE (default).

        int: flag for intercept; if TRUE (default) an intercept
             term is included in the regression.

        tol: tolerance parameter for rq computations.

         ci: flag for confidence interval; if TRUE (default)
             the confidence intervals are returned.

     method: if method="score" (default), ci is computed using
             regression rank score inversion; if method="spar-
             sity", ci is computed using sparsity function.

   interpolate: if TRUE (default), the smoothed confidence
             intervals are returned.

      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.

         hs: logical flag to use Hall-Sheather's sparsity esti-
             mator (default); otherwise Bofinger's version is
             used.

   DDeessccrriippttiioonn::

        Perform a quantile regression on a design matrix, x, of
        explanatory variables and a vector, y, of responses.

   VVaalluuee::

       coef: the estimated parameters of the tau-th conditional
             quantile function.

      resid: the estimated residuals of the tau-th conditional
             quantile function.

       dual: the dual solution (if dual=T).

          h: the index of observations in the basis.

         ci: confidence intervals (if ci=T).

   MMEETTHHOODD::

        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).

   SSEEEE AALLSSOO::

        trq and qrq for further details and references.

   AAuutthhoorr((ss))::

        Roger Koenker, roger@ysidro.econ.uiuc.edu, <URL:
        http://www.econ.uiuc.edu/~roger/research/rq/rq.html>.
        Ported to R, and added rq.formula, by Kjetil Halvorsen.

   RReeffeerreenncceess::

        [1] Koenker, R.W. and Bassett, G.W. (1978). Regression
        quantiles, Econometrica, 46, 33-50.

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

        [3] Gutenbrunner, C. Jureckova, J. (1991).  Regression
        quantile and regression rank score process in the lin-
        ear 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-Ver-
        lag, New York.

   EExxaammpplleess::

        data(stackloss)
        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
        data(Patacamaya)               # an example with formula
         z0.1 <- rq.formula(y ~ a+tipo, data=Patacamaya, na.action=na.omit, tau=0.1)
        z0.1$coef
        z0.1$ci

