Likelihood, score function and information matrix, approximate ancillary statistics and sample space derivative for the generalized Pareto distribution parametrized in terms of return levels.
vector of length 2 containing \(y_m\) and \(\xi\), respectively the \(m\)-year return level 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 gpdr.Vfun
gpdr.ll(par, dat, m, tol=1e-5)
gpdr.ll.optim(par, dat, m, tol=1e-5)
gpdr.score(par, dat, m)
gpdr.infomat(par, dat, m, method = c('obs', 'exp'), nobs = length(dat))
gpdr.Vfun(par, dat, m)
gpdr.phi(par, V, dat, m)
gpdr.dphi(par, V, dat, m)
gpdr.ll: log likelihood
gpdr.ll.optim: negative log likelihood parametrized in terms of log(scale) and shape
in order to perform unconstrained optimization
gpdr.score: score vector
gpdr.infomat: observed information matrix for GPD parametrized in terms of rate of \(m\)-year return level and shape
gpdr.Vfun: vector implementing conditioning on approximate ancillary statistics for the TEM
gpdr.phi: canonical parameter in the local exponential family approximation
gpdr.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.
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.