[RsR] The robust regression confidence interval‏

[Anton Kochepasov]

Hi everyone,

A few years ago there was a discussion about a robust regression confidence interval (https://stat.ethz.ch/pipermail/r-sig-robust/2008/000217.html) and I would like to resort your courtesy again.

I'm trying to compare a few regression models for my data. For linear regression everything is quite understandable, but robust and quantile regressions are not so obvious. I could not find almost anything about calculating confidence interval for these regression models unless I looked for something wrong.

My code in R looks as follows:
# Robust linear modeling
library(MASS)
library(robustbase)
library(robust)
set.seed(343); 
x <- rnorm(1000)
y <- x + 2*rnorm(1000)

lm1<-lm(y~x); rlm1<-rlm(y~x); rlm2 <- lmRob(y~x); rlm3 <- lmrob(y~x)
cbind(summary(lm1)$coeff,  confint(lm1)) 
cbind(summary(rlm1)$coeff, confint(rlm1))
cbind(summary(rlm2)$coeff, confint(rlm2))
cbind(summary(rlm3)$coeff, confint(rlm3))

And produces the following result:
> cbind(summary(lm1)$coeff,  confint(lm1))
               Estimate Std. Error   t value     Pr(>|t|)      2.5 %     97.5 %
(Intercept) -0.06973191 0.06408983 -1.088034 2.768429e-01 -0.1954982 0.05603438
x            0.97647196 0.06619635 14.751145 1.071805e-44  0.8465720 1.10637196
> cbind(summary(rlm1)$coeff, confint(rlm1))
                  Value Std. Error    t value 2.5 % 97.5 %
(Intercept) -0.06131788 0.06714405 -0.9132288    NA     NA
x            0.96016596 0.06935096 13.8450275    NA     NA
> cbind(summary(rlm2)$coeff, confint(rlm2))
Error in UseMethod("vcov") : 
  no applicable method for 'vcov' applied to an object of class "lmRob"
> cbind(summary(rlm3)$coeff, confint(rlm3))
              Estimate Std. Error    t value     Pr(>|t|)      2.5 %     97.5 %
(Intercept) -0.0568964 0.06608987 -0.8608945 3.895029e-01 -0.1865874 0.07279464
x            0.9612520 0.06821558 14.0913850 2.921913e-41  0.8273896 1.09511448

It's easy to spot that linear model works OK and only one robust regression gives a sensible result. Another observation is that lmrob(), which produces some actual confidence interval, calculates it in the same manner as lm(), with using 1.96 as the student coefficient.

Could you share your opinion if it is a correct way to produce a confidence interval for the robust regression model (same as for the linear regression)? May the same method be used for the quantile regression model? If not, what should I use?

Thank you in advance,
Anton