Tangent exponential model approximation for the GP distribution

The function gpd.tem provides a tangent exponential model (TEM) approximation for higher order likelihood inference for a scalar parameter for the generalized Pareto distribution. Options include scale and shape parameters as well as value-at-risk (also referred to as quantiles, or return levels) and expected shortfall. The function attempts to find good values for psi that will cover the range of options, but the fit may fail and return an error. In such cases, the user can try to find good grid of starting values and provide them to the routine.

Usage

gpd.tem(
  dat,
  param = c("scale", "shape", "quant", "VaR", "retlev", "ES", "Nmean", "Nquant"),
  psi = NULL,
  m = NULL,
  thresh = 0,
  n.psi = 50,
  N = NULL,
  p = NULL,
  q = NULL,
  plot = FALSE,
  correction = TRUE,
  ...
)

Arguments

dat: sample vector for the GP distribution
param: parameter over which to profile
psi: scalar or ordered vector of values for the interest parameter. If NULL (default), a grid of values centered at the MLE is selected. If psi is of length 2 and n.psi>2, it is assumed to be the minimal and maximal values at which to evaluate the profile log likelihood.
m: number of observations of interest for return levels. See Details. Required only for param = 'VaR' or param = 'ES'.
thresh: threshold value corresponding to the lower bound of the support or the location parameter of the generalized Pareto distribution.
n.psi: number of values of psi at which the likelihood is computed, if psi is not supplied (NULL). Odd values are more prone to give rise to numerical instabilities near the MLE
N: size of block over which to take maxima. Required only for args Nmean and Nquant.
p: tail probability, equivalent to \(1/m\). Required only for args quant.
q: level of quantile for N-block maxima. Required only for args Nquant.
plot: logical indicating whether plot.fr should be called upon exit
correction: logical indicating whether spline.corr should be called.
...: additional arguments, for backward compatibility

Value

an invisible object of class fr (see tem in package hoa) with elements

normal: maximum likelihood estimate and standard error of the interest parameter \(\psi\)
par.hat: maximum likelihood estimates
par.hat.se: standard errors of maximum likelihood estimates
th.rest: estimated maximum profile likelihood at (\(\psi\), \(\hat{\lambda}\))
r: values of likelihood root corresponding to \(\psi\)
psi: vector of interest parameter
q: vector of likelihood modifications
rstar: modified likelihood root vector
rstar.old: uncorrected modified likelihood root vector
param: parameter

Details

As of version 1.11, this function is a wrapper around gpd.pll.

The interpretation for m is as follows: if there are on average \(m_y\) observations per year above the threshold, then \(m = Tm_y\) corresponds to \(T\)-year return level.

Author

Leo Belzile

Examples

set.seed(123)
dat <- rgp(n = 40, scale = 1, shape = -0.1)
#with plots
m1 <- gpd.tem(param = 'shape', n.psi = 50, dat = dat, plot = TRUE)


if (FALSE) { # \dontrun{
m2 <- gpd.tem(param = 'scale', n.psi = 50, dat = dat)
m3 <- gpd.tem(param = 'VaR', n.psi = 50, dat = dat, m = 100)
#Providing psi
psi <- c(seq(2, 5, length = 15), seq(5, 35, length = 45))
m4 <- gpd.tem(param = 'ES', dat = dat, m = 100, psi = psi, correction = FALSE)
mev:::plot.fr(m4, which = c(2, 4))
plot(fr4 <- spline.corr(m4))
confint(m1)
confint(m4, parm = 2, warn = FALSE)
m5 <- gpd.tem(param = 'Nmean', dat = dat, N = 100, psi = psi, correction = FALSE)
m6 <- gpd.tem(param = 'Nquant', dat = dat, N = 100, q = 0.7, correction = FALSE)
} # }