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I need the description of Huber sandwich estimate method for quantile regression.

I found this "a Huber sandwich estimate using a local estimate of the sparsity function". Sparsity function looks like $s(\theta)=f(F^{-1}(\theta))^{-1}$, $F$ is a distribution of residuals and $f$ is a density function of $F$. So my question is what exactly means "local estimate of sparsity function":)

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  • $\begingroup$ At the moment this reads like a request to do homework. If that's not the case, perhaps a description of what level of explanation is needed and what sort of searching efforts you have already exerted would increase the probability people to engage with you. $\endgroup$
    – DWin
    Commented Dec 14, 2013 at 17:24
  • $\begingroup$ @Dwin thank you for your answer, but for me is still not clear what when is the difference between sparsity method (which is used when residuals are iid) and Huber sandwich method (when residuals are non-iid). $\endgroup$
    – justep
    Commented Dec 15, 2013 at 15:34
  • $\begingroup$ I didn't mean to suggest that there were two different methods; only that there was a theoretic aspect being raised in the SAS docs and a computational aspect that was better addressed in Koenker's R documentation. $\endgroup$
    – DWin
    Commented Dec 15, 2013 at 17:09
  • $\begingroup$ What I am looking for is the difference between these to methods in theoretic way. So that's why I was asking for description of Huber sandwich method, because sparsity method looks quite clear, but I don't understand what exactly changes are made in Huber sandwich case that it is already possible to use non-iid residuals. But anyway, thanks;) $\endgroup$
    – justep
    Commented Dec 15, 2013 at 17:37
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    $\begingroup$ "This estimator of the sparsity function is sensitive to the iid assumption. Alternately, Koenker and Machado (1999) considered the non-iid case. By assuming local linearity <..>, they proposed a local estimator of the density function by using the difference quotient. A Huber sandwich estimate of the covariance <..> used to construct the confidence intervals." This is from SAS QUANTREG manual section. So as I understand there are two (or more) ways to estimate this sparsity function and one is for iid residuals and another for non-iid residuals (Huber sand). Question - what is the difference? $\endgroup$
    – justep
    Commented Dec 15, 2013 at 17:55

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That appears to be from the SAS QUANTREG manual section: They then go on to call the sparsity function. the "reciprocal of the density function" (which is arguably a natural language version of the formula above but is more suggestive that a local sparsity would be something like 1/(r_1-r_2) where r-1 and r_2 are local density residuals rather than just a functional "inverse".) So a natural interpretation is that the (local) sparsity function is very similar to the average distance between nearby ordered residuals. In fact this formula for empiric estimate of s() appears immediately below that description:

enter image description here

So the sparsity function is the local estimate of the average distance between nearby residuals on the original scale in the limit as "nearby" (defined by the bandwidth) gets small. I didn't really see how that matched up with what I knew about sandwich estimators, so I also looked at Koenker's quantreg R package where the mathematical definition appears:

> Hinv: inverse of the estimated Hessian matrix returned if cov=TRUE and
        se != "iid". 
> The Huber sandwich is cov = Hinv %*% J %*% Hinv.

The flanking Hinv matrices are what give the name "sandwich" to the method. J is sandwiched between two Hinv's.

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