1\name{rq.fit.hogg}
2\alias{rq.fit.hogg}
3\title{weighted quantile regression fitting}
4\description{
5Function to estimate a regression mmodel by minimizing the weighted sum of several
6quantile regression functions.  See Koenker(1984) for an asymptotic look at these
7estimators.  This is a slightly generalized version of what Zou and Yuan (2008) call
8composite quantile regression in that it permits weighting of the components of the
9objective function and also allows further linear inequality constraints on the coefficients.
10}
11\usage{
12rq.fit.hogg(x, y, taus = c(0.1, 0.3, 0.5), weights = c(0.7, 0.2, 0.1),
13	    R = NULL, r = NULL, beta = 0.99995, eps = 1e-06)
14}
15\arguments{
16  \item{x}{design matrix}
17  \item{y}{response vector }
18  \item{taus}{quantiles getting positive weight}
19  \item{weights}{weights assigned to the quantiles }
20  \item{R}{optional matrix describing linear inequality constraints}
21  \item{r}{optional vector describing linear inequality constraints}
22  \item{beta}{step length parameter of the Frisch Newton Algorithm}
23  \item{eps}{tolerance parameter for the Frisch Newton Algorithm}
24}
25\details{
26Mimimizes a weighted sum of quantile regression objective functions using
27the specified taus.  The model permits distinct intercept parameters at
28each of the specified taus, but the slope parameters are constrained to
29be the same for all taus.  This estimator was originally suggested to
30the author by Bob Hogg in one of his famous blue notes of 1979.
31The algorithm used to solve the resulting linear programming problems
32is either the Frisch Newton algorithm described in Portnoy and Koenker (1997),
33or the closely related algorithm described in Koenker and Ng(2002) that
34handles linear inequality constraints.  See \code{\link{qrisk}} for illustration
35of its use in portfolio allocation.
36
37Linear inequality constraints of the form \eqn{Rb \geq r} can be imposed  with
38the convention that \eqn{b} is a \eqn{m+p} where \eqn{m} is the \code{length(taus)}
39and \eqn{p} is the column dimension of \code{x} without the intercept.
40}
41\value{
42  \item{coefficients}{estimated coefficients of the model}
43}
44\references{
45Zou, Hui and and Ming Yuan (2008)  Composite quantile regression and the
46Oracle model selection theory, Annals of Statistics, 36, 1108--11120.
47
48Koenker, R. (1984) A note on L-estimates for linear models,
49Stat. and Prob Letters, 2, 323-5.
50
51Portnoy, S. and Koenker, R. (1997) The Gaussian Hare and the
52Laplacean Tortoise:  Computability of Squared-error vs Absolute Error Estimators,
53(with discussion).  Statistical Science, (1997) 12, 279-300.
54
55Koenker, R. and Ng, P (2003) Inequality Constrained Quantile Regression, preprint.
56 }
57\author{ Roger Koenker }
58
59
60\seealso{ \code{\link{qrisk}}}
61
62\keyword{regression}
63\keyword{ robust }
64