Quantile Regression (rq)


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

USAGE:
       rq(x, y, tau=-1, alpha=.1, dual=F, int=T, tol=1e-4, ci = T, method="score",
       interpolate=T, tcrit=T, hs=T)


REQUIRED ARGUMENTS:
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.

OPTIONAL ARGUMENTS:
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 intervals; de-
       fault 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 confi-
       dence intervals are returned.
method:  if method="score" (default), ci is  computed  using  re-
       gression rank score inversion; if method="sparsity", ci is
       computed using sparsity function.
interpolate:  if TRUE (default), the smoothed  confidence  inter-
       vals 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 estimator
       (default); otherwise Bofinger's version is used.

VALUE:
coef:  the estimated parameters of the tau-th  conditional  quan-
       tile function.
resid:   the  estimated residuals of the tau-th conditional quan-
       tile function.
dual:  the dual solution (if dual=T).
h:     the index of observations in the basis.
ci:    confidence intervals (if ci=T).

VALUE:
sol:   a  (p+2) by m matrix whose first row contains the  'break-
       points'  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.
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 interpreta-
       tion of dsol.
h:     the matrix of observations indices  in  the  basis  corre-
       sponding to sol or dsol.

EXAMPLES:
       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

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).

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 Regres-
       sion  Quantiles, in P. Mandl and M. Huskova (eds.), Asymp-
       totic Statistics, 349-359, Springer-Verlag, New York.


SEE ALSO:
       trq and qrq for further details and references.






Linearized Quantile Estimation (qrq)


DESCRIPTION:
       Compute linearized quantiles from rq data structure.

USAGE:
       qrq(s, a)

REQUIRED ARGUMENTS:
s:      data  structure returned by the quantile regression func-
       tion rq with t<0 or t>1.
a:     the vector of quantiles for which the  corresponding  lin-
       earized quantiles are to be computed.

VALUE:
       a vector of the linearized quantiles corresponding to vec-
       tor a, as interpolated from the second row of s$sol.

SEE ALSO:
       rq and  trq  for further detail.

EXAMPLES:
       z_qrq(rq(x,y),a)       #assigns z the linearized quantiles
                              #corresponding to vector a.








Function  to compute analogues of the trimmed mean for the linear
regression model. (trq)


DESCRIPTION:
       The function returns a regression trimmed  mean  and  some
       associated  test statistics.  The proportion a1 is trimmed
       from the lower tail  and  a2  from  the  upper  tail.   If
       a1+a2=1 then a result is returned for the a1 quantile.  If
       a1+a2<1 two methods of trimming are possible described be-
       low  as  "primal" and "dual". The function "trq.print" may
       be used to print results in the style of ls.print.

USAGE:
       trq(x, y, a1=0.1, a2,  int=T, z,  method="primal", tol=1e-4)

REQUIRED ARGUMENTS:
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  elemants  of   y,
       and  there   should   be fewer columns than rows.  Missing
       values are not  allowed.
y:     reponse vector with as many observations as the number  of
       rows of x.  Missing value are not allowed.

OPTIONAL ARGUMENTS:
a1:     the lower trimming proportion; defaults to .1 if missing.
a2:    the upper trimming proportion; defaults to a1 if  missing.
int:   flag for intercept; if TRUE, an intercept term is included
       in regrssion model.  The  default  includes  an  intercept
       term.
z:      structure  returned by the function 'rq' with t <0 or >1.
       If missing, the function rq(x,y,int=int) is  automatically
       called to generate this argument.  If several calls to trq
       are anticipated for the same data this avoids  recomputing
       the rq solution for each call.
method:   method  to  be used for the trimming.  If the choice is
       "primal", as is the default, a trimmed mean of the  primal
       regression  quantiles   is computed based on the sol array
       in the 'rq' structure.  If the method is "dual", a weight-
       ed  least-squares  fit  is done using the dual solution in
       the 'rq'  structure  to  construct  weights.   The  former
       method  is discussed in detail in Koenker and Potnoy(1987)
       the latter in Ruppert and Carroll(1980)  and  Gutenbrunner
       and Jureckova(1991).
tol:   Tolerance parameter for rq computions

VALUE:
coef:  estimated coeficient vector
resid:  residuals from the fit.
cov:   the estimated covariance matrix for the coeficient vector.
v:     the scaling factor of the covariance matrix under iid  er-
       ror assumption: cov=v*(x'x)^(-1).
wt:     the  weights  used in the least squares computation,  Re-
       turned only when method="dual".
d:     the bandwidth used to compute the sparsity function.   Re-
       turned only when a1+a2=1.

EXAMPLES:
       z_rq(x,y)  #z gets the full regression quantile structure
       trq(x,y,.05,z=z)  #5% symmetric primal trimming
       trq(x,y,.01,.03,method="dual")  #1% lower and 3% upper trimmed least-
                                    #squares fit.
       trq.print(trq(x,y)) #prints trq results in the style of ls.print.

METHOD:
       details  of  the methods may be found in Koenker and Port-
       noy(1987) for the case of primal trimming  and  in  Guten-
       brunner and Jureckova(1991) for dual trimming.  On the es-
       timation of the covariance  matrix  for  individual  quan-
       tiles,  see  Koenker(1987) and the discussion in Hendricks
       and Koenker(1991).  The estimation of the  covariance  ma-
       trix  under   non-iid conditions is an open research prob-
       lem.

REFERENCE:
       Bassett, G., and Koenker, R. (1982), "An  Empirical  Quan-
       tile  Function for Linear Models With iid Errors," Journal
       of the American Statistical Association, 77, 407-415.

       Koenker, R.W. (1987), "A Comparison of Asymptotic  Methods
       of  Testing  based  on  L1  Estimation," in Y. Dodge (ed.)
       Statistical Data Analysis Based on the L1 norm and Related
       Methods, New York:  North-Holland.

       Koenker, R. W., and Bassett, G.W (1978), "Regression Quan-
       tiles", Econometrica, 46, 33-50.

       Koenker, R., and Portnoy,  S.  (1987),  "L-Estimation  for
       Linear  Models", Journal of the American Statistical Asso-
       ciation, 82, 851-857.

       Ruppert, D.  and  Carroll,  R.J.  (1980),  "Trimmed  Least
       Squares  Estimation  in  the Linear Model", Journal of the
       American Statistical Association, 75, 828-838.
