[RsR] Robust location estimator - an interesting application in finance

[Andreas Ruckstuhl]

Dear Ajay

I agree on what Matias said.

In addition, I wondered whether you should not better use a "direct" 
robust location estimator instead of the robust location estimator based 
on the regression case. For example, there is R function huberM() which 
returns a robust M-estimation of location based on a safe way of 
calculating the robust scale estimator. An additional advantage of using 
huberM() is, that the resulting M-estimator has breakdown-point 0.5 
which is (much) higher than that of a trimmed mean.

To sell the robust M-estimator, you can talk of "an estimator which 
down-weights outliers automatically according to their outlyingness".
(The M-estimator can also be modified that it removes very distant 
outliers completely by using a redescending psi-function.)

All the best
Andreas


Some addtional comments

Ajay Shah schrieb:
> One interesting application of a robust location estimator is in
> computing reference rates on OTC markets. Traders on an OTC market
> know the ruling price but others do not. So an information agency asks
> a bunch of dealers what the price is.
> 
> Dealers typically have positions on the market and have an incentive
> to lie. Hence, it's useful to have a robust location estimator. The
> British Bankers Association has used a `fixed trimmed mean' where the
> four most extreme observations are thrown away and the average of the
> remainder is used as the `reference rate' of the market. This is the
> method underlying LIBOR.
> 
> A while ago, Donald Lien and John Cita suggested that it would make
> more sense to experiment with a few different levels of trimming, and
> pick the one where the standard deviation of the trimmed mean
> (obtained through the bootstrap) is the lowest. They termed this the
> `adaptive trimmed mean' or the ATM.
> 
> One advantage of the above two ideas is that they are simple to
> explain to regulators and traders.
> 
> My question is: How far can contemporary knowledge in robust
> statistics improve upon this scheme? If one uses robustbase::lmrob(x ~
> 1) and gets a location estimator, would it be much better?
> 
> Here is some data for experimentation:
> 
>   load(url("http://www.mayin.org/ajayshah/tmp/all.rda"))
> 
> This gives you an object "all" which has 44 columns of data. Each of
> these columns is one set of values obtained from a bunch of dealers.
> 
> I did:
> 
>   library(refrate)
>   results <- matrix(NA, nrow=length(fileslist), ncol=4)
>   colnames(results) <- c("lmrob","median","atm","mean")
>   for (i in 1:length(fileslist)) {
>     tmp <- na.omit(all[,i])
>     a <- try(lmrob(tmp ~ 1)$coefficients)
>     result <- NA
>     if (class(a) != "try-error") {result <- a}
>     results[i,] <- c(result,
>                      median(tmp),
>                      referencerate(tmp)["atm"],
>                      mean(tmp))
>   }
>   cor(results, use="pairwise.complete.obs")
> 
> where the function referencerate() implements the Lien/Cita scheme
> described above. (I can email you this code if there is interest). I
> have two findings:
> 
> (a) lmrob() often breaks. It shouldn't. I have sent in one bug report.
> 
> (b) The correlation matrix shows very high correlations:
> 
>              lmrob    median       atm      mean
>   lmrob  1.0000000 
>   median 0.9998192 1.0000000 
>   atm    0.9999741 0.9998113 1.0000000 
>   mean   0.9993983 0.9994536 0.9996133 1.0000000
> 
> The correlations with the ATM are: lmrob > median > mean. So lmrob()
> and the ATM seem to agree a lot.
> 
> Looking deeper, an important feature in this (financial) application
> is that dealers should not see a location estimator where a small
> cartel can produce a large distort the price. So their gains from
> forming a cartel should be low. Would lmrob() be much different from
> the ATM in this?
> 

-- 
----------------------------------------------------------------------

Prof. Dr. Andreas Ruckstuhl
ZHAW Zürcher Hochschule für Angewandte Wissenschaften
IDP Institut für Datenanalyse und Prozessdesign
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