Processing math: 100%
Skip to contents

Likelihood, score function and information matrix, approximate ancillary statistics and sample space derivative for the generalized Pareto distribution parametrized in terms of expected shortfall.

The parameter m corresponds to ζu/(1-α), where ζu is the rate of exceedance over the threshold u and α is the percentile of the expected shortfall. Note that the actual parametrization is in terms of excess expected shortfall, meaning expected shortfall minus threshold.

Arguments

par

vector of length 2 containing em and ξ, respectively the expected shortfall at probability 1/(1-α) and the shape parameter.

dat

sample vector

m

number of observations of interest for return levels. See Details

tol

numerical tolerance for the exponential model

method

string indicating whether to use the expected ('exp') or the observed ('obs' - the default) information matrix.

nobs

number of observations

V

vector calculated by gpde.Vfun

Details

The observed information matrix was calculated from the Hessian using symbolic calculus in Sage.

Usage

gpde.ll(par, dat, m, tol=1e-5)
gpde.ll.optim(par, dat, m, tol=1e-5)
gpde.score(par, dat, m)
gpde.infomat(par, dat, m, method = c('obs', 'exp'), nobs = length(dat))
gpde.Vfun(par, dat, m)
gpde.phi(par, dat, V, m)
gpde.dphi(par, dat, V, m)

Functions

  • gpde.ll: log likelihood

  • gpde.ll.optim: negative log likelihood parametrized in terms of log expected shortfall and shape in order to perform unconstrained optimization

  • gpde.score: score vector

  • gpde.infomat: observed information matrix for GPD parametrized in terms of rate of expected shortfall and shape

  • gpde.Vfun: vector implementing conditioning on approximate ancillary statistics for the TEM

  • gpde.phi: canonical parameter in the local exponential family approximation

  • gpde.dphi: derivative matrix of the canonical parameter in the local exponential family approximation

Author

Leo Belzile