rq

0th

Percentile

Quantile Regression

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

Keywords
regression
Usage
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)
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
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 intervals; 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="sparsity", 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 estimator (default); otherwise Bofinger's version is used.
Value

• sola (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.

• dsol
• h

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).

trq and qrq for further details and references.

References

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

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

 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.

 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.

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

• rq
• rq.formula
Examples
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
Documentation reproduced from package quantreg, version 2.0-2, License: This software has been submitted to Statlib and may be freely used and redistributed for non-commercial purposes. See Readme file.

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