# Prediction interval for robust regression with MM-estimator

In their book "Robust Statistics", Maronna et al. consider the following model for robust regression: $y_i = \beta x_i + u_i$, where $u_i$ are independent of the $x_i$, and are i.i.d, with finite variance. They go on to provide a robust estimate $\hat{\beta}$ of $\beta$ which is asymptotically normal and give the covariance matrix for $\hat{\beta}$. My question is that without any knowledge about the distribution of $u_i$, is it possible to provide a prediction interval for $y$ (without using bootstrap)? I'm asking this because

library(MASS)
robustModel = rlm(formula = myFormula, data = myData, method = "MM")
predict.rlm(robustModel, newdata = myNewData, interval = "prediction")


in R generates a prediction interval. For reference, this is the code for predict.rlm:

predict.rlm <- function (object, newdata = NULL, scale = NULL, ...)
{
## problems with using predict.lm are the scale and
## the QR decomp which has been done on down-weighted values.
object$qr <- qr(sqrt(object$weights) * object$x) predict.lm(object, newdata = newdata, scale = object$s, ...)
}


It seems to me that the prediction interval that is obtained this way is for normally distributed $u_i$. Is that correct? What am I missing here?

• thank you for response. The estimator is obtained using an iterative procedure, so there's really no closed form for it. I can understand your comment about the confidence interval: we're looking for a confidence interval for $\hat{\beta} x$ (assume 1-d) and $\hat{\beta}$ is asymptotically normal, and $\Var(\hat{\beta}x) =$x^2Var(\hat{\beta})$and from here we can get the confidence interval for$\hat{\beta}x$. But without any idea about the distribution of$u$, I can't see how to get the a (approx) confidence interval for$\hat{\beta} x + u$(i.e. a prediction interval). – user765195 Oct 2 '12 at 1:57 • Thank you for response @michaelchernick. The estimator is obtained using an iterative procedure, so there's really no closed form for it. I can understand your comment about the confidence interval: we're looking for a confidence interval for$\hat{\beta} x$(assume 1-d) and$\hat{\beta}$is asymptotically normal, and$Var(\hat{\beta}x) = x^2Var(\hat{\beta})$and from here we can get the confidence interval for$\hat{\beta}x$. But without any idea about the distribution of$u$, I can't see how to get the a (approx) confidence interval for$\hat{\beta} x + u\$ (i.e. a prediction interval). – user765195 Oct 2 '12 at 2:03