[R-SIG-Finance] Combining a kernel density interior + GPD tailed CDF

[Gareth McEwan]

Hi all

Sorry for the double post, I needed to add a heading..

I need to transform standardized GARCH residuals to uniform variates by a
marginal semi-parametric empirical CDF.

For a set of 242 standardized residuals (sorted.ALBI.std.resids), I have
cut off 10% at the upper tail and 10% at the lower tail (by theory and
testing, these thresholds are valid).

I have a set of 24 (positive) values for the lower tail (by multiplying by
-1), a set of 24 values for the upper tail, and 194 values for the interior
(kernel) of the distribution.

These values correspond to a choice of threshold at
quantile(neg.sorted.ALBI.resids,0.90)  for the lower tail #equals 1.213646

and

quantile(sorted.ALBI.std.resids,0.90) for the upper tail #equals  1.218522.

I used the "evir", "ismev", and "fExtremes" packages to fit and
double-check estimates and they are all extremely close.

Output:

For lower tail:
gpdfit.mle.ALBI =
gpdFit(neg.tail.ALBI.resids,u=quantile(neg.sorted.ALBI.resids,0.90),type="mle")
summary(gpdfit.mle.ALBI)
#       xi (shape)           beta (scale)        LLH Value
#       0.1051834          0.7098641          18.30353
#SE: 0.2443332    SE: 0.2254843

For upper tail:
gpdfit.mle.ALBI =
gpdFit(pos.tail.ALBI.resids,u=quantile(sorted.ALBI.std.resids,0.90),type="mle")
summary(gpdfit.mle.ALBI) # Maximum likelihood parameter estimation method
#        xi (shape)            beta (scale)       LLH Value
#       -0.1996078           0.4973661         2.447625
#SE:   0.2571947    SE: 0.1620495

For interior:
fit.interior.ALBI.std.resids =
density(kernel.ALBI.std.resids,bw="nrd0",adjust=1,kernel="gaussian")

The question I'd like to ask is: how do I now combine the non-parametric
interior with the estimated tails to get a CDF function? It is this
semi-parametric CDF function that I need to use to transform the
standardized residuals to uniform variates.

Thank you very much for the help (files are attached for reproducibility).

Best regards
Gareth

P.S.  I would prefer to use the above framework, if possible. However, I
did use the following, but got conflicting output for the threshold and
tail estimates:
spd.fit.ALBI=spdfit(sorted.ALBI.std.resids, upper=0.9, lower=0.1,
tailfit="GPD", type="mle", kernelfit="normal", information="observed")
show(spd.fit.ALBI)
# Upper Tail: Threshold = 1.22226 (compared to 1.218522 using quantile
function)
# Estimated Parameters
#     xi           beta
#   -0.18608   0.48698
# Lower Tail: Threshold = -1.22126 (compared to -1.213646 using quantile
function on sorted.ALBI.std.resids)
# Estimated Parameters
#     xi           beta
#    0.06574   0.76526
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