A standard idea in extreme value theory (see e.g. here, in French unfortunately) is that to estimate the 99.5% quantile (say), we just need to estimate a quantile of level 95% for observations exceeding the 90% quantile.

In extreme value theory, we assume that the 90% quantile (of the initial distribution) can be obtained easily, e.g. the empirical quantile, and then, for the exceeding observations, we fit a Pareto distribution (a Generalized Pareto one to be precise), and get a parametric quantile for the 95% quantile. I.e.

which can be written
So, an estimation of the cumulative distribution function is
and if we invert it, we get the popular expression for high level quantiles,
Hence, we do not really care about observations in the core of the distribution.

And I was wondering if this can be transposed with quantile regressions. Hence, I would like to get a quantile regression of level 90% (say) of given , based on observations ‘s, but all observations such that for some are missing. More precisely, I have the following sample (here half of the observations are missing),

Assume that we know that I have observations below the quantile of level 25%, and above the quantile of level 75%.
If I want to get the 90% quantile regression, and the 10% quantile, the code is simply,

library(mnormt)
library(quantreg)
library(splines)
set.seed(1)
mu=c(0,0)
r=0
Sigma <- matrix(c(1,r,r,1), 2, 2)
Z=rmnorm(2500,mu,Sigma)
X=Z[,1]
Y=Z[,2]
 
base=data.frame(X,Y)
plot(X,Y,col="blue",cex=.7)
I=(Y>qnorm(.25))&(Y
The graph is the following


Dotted lines - in black - are theoretical lines (if I had all
observations), and plain lines are (where half of the sample if
missing). Instead of a standard linear quantile regression, it is also
possible to try a spline regression,


So obviously, if I miss something in the middle, that's no big deal, doted and plain lines are here extremely close.

But what if observations  and

were correlated ? Consider a Gaussian random vector  with
correlation 
(here 0.6).


It looks like we overestimate the slope for high quantile, but not for
lower quantiles. So if observations are correlated, we have to be
cautious with that technique.

But why could that be interesting ? Well, because I wanted to run a
quantile regression on marathon results. But I could not get the
overall dataset (since I had to import observations manually, and I have to
admit that it was a bit boring). So I extracted finish times of the first
10% athletes, and the latest 10%. And I was
wondering if it was enough to look at the 5% and 95% quantiles, based
on the age of the runner... To be
continued.

		

	
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