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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