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.
Arguments
- par
vector of length 2 containing \(e_m\) and \(\xi\), respectively the expected shortfall at probability 1/(1-\(\alpha\)) 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 likelihoodgpde.ll.optim: negative log likelihood parametrized in terms of log expected shortfall and shape in order to perform unconstrained optimizationgpde.score: score vectorgpde.infomat: observed information matrix for GPD parametrized in terms of rate of expected shortfall and shapegpde.Vfun: vector implementing conditioning on approximate ancillary statistics for the TEMgpde.phi: canonical parameter in the local exponential family approximationgpde.dphi: derivative matrix of the canonical parameter in the local exponential family approximation