Hi all 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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