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 \(\zeta_u\)/(1-\(\alpha\)), where \(\zeta_u\) is the rate of exceedance over the threshold
u and \(\alpha\) is the percentile of the expected shortfall.
Note that the actual parametrization is in terms of excess expected shortfall, meaning expected shortfall minus threshold.
vector of length 2 containing \(e_m\) and \(\xi\), respectively the expected shortfall at probability 1/(1-\(\alpha\)) and the shape parameter.
sample vector
number of observations of interest for return levels. See Details
numerical tolerance for the exponential model
string indicating whether to use the expected ('exp') or the observed ('obs' - the default) information matrix.
number of observations
vector calculated by gpde.Vfun
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)
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
Leo Belzile
The observed information matrix was calculated from the Hessian using symbolic calculus in Sage.