gpd.tail: POT ESTIMATION OF A GENERALIZED PARETO DISTRIBUTION
In Rsafd: Statistical Analysis of Financial Data in R

Description Usage Arguments Value Note Author(s) See Also Examples

Semi-parametric estimation based on the Peaks over Threshold (POT) method.of a distribution with upper and possibly lower tail have the same size as the tail of a generalized Pareto distribution.

1 2	gpd.tail(data, one.tail=F, upper = NA, lower = NA, upper.method ="ml", lower.method="ml", plot = T, ...)

`data`	A numerical vector of sample data values
`Optional Arguments`
`one.tail`	Logical which should be set to `TRUE` when the sample is limited to the left
`upper`	The value of the upper threshold. A Generalized Pareto distribution is fitted to the data exceeding this threshold. If its value is missing, `upper` is estimated so that either 15% of the data samples, or exactly 150 values, whichever number is greater, are greater than the upper threshold.
`lower`	The value of the lower threshold. A Generalized Pareto distribution is fitted to the data below this threshold. If its value is missing, it is estimated so that either 15% of the data samples, or exactly 150 values, whichever number is smaller, is smaller than the lower threshold.
`upper.method`	Character string which can be either "ml" or "lmom" depending upon the choice of the method used to estimate the parameters of the generalized Pareto distribution fitted to the part of the data exceeding `upper`
`lower.method`	Character string which can be either "ml" or "lmom" depending upon the choice of the method used to estimate the parameters of the generalized Pareto distribution fitted to the part of the data below `lower`
`plot`	If `TRUE` (default), for each tail, a plot of the excesses over threshold against the quantiles of the GPD with the same shape parameter is produced. The fact that the points line up supports the assumption about the (generalized) Pareto nature of the tail in question.

The value of gpd.tail is an object of class gpd. It is a list containing fields: length of the data, sorted vector of data points, the value of upper threshold, and estimates of GPD shape and scale parameter for the tail. and representing threshold and parameter estimates for the lower tail.

If plot = TRUE a plot of the exceedences over the threshold against the quantiles of the estimated GPD is produced for each of the tails studied.

The semiparametric estimation performed by this function was done by the functions pot.1tail.est and pot.2tails.est in the original EVANESCE library. These functions were included in the FinMetrics library of S-Plus under the name of gpd.tail to avoid confusion with the similar function gpd contributed by Mc Neil. We named it gpd.tail for backward compatibility with FinMetrics.

Rene Carmona, rcarmona@princeton.edu

gpd.lm, gpd.lmom

# One tail
data("BCofLRet")
NZ <- (BCofLRet !=0)
BLRet <- BCofLRet[NZ]
X <- BLRet[BLRet > 0]
x.est <- gpd.tail(X,one.tail=TRUE,upper=0.04)

y <- c(10:500)/1000
plot(y, gpd.1p(y,x.est), log = "x", type = "l")
big.X <- sort(X) > 0.01
points((sort(X))[big.X], (ppoints(sort(X)))[big.X])

# random generation from this distribution:
X <- BLRet[BLRet != 0]
x.est <- gpd.tail(X, upper = 0.015, lower = -0.015, method = "lmom")
n <- length(X)
Y <- gpd.2q(runif(n), x.est)
plot(X, ylim = c(-0.3,0.3))
plot(Y, col = 4, ylim = c(-0.3,0.3))