Likelihood, score function and information matrix,
approximate ancillary statistics and sample space derivative
for the generalized Pareto distribution parametrized in terms of average maximum of N exceedances.
The parameter N corresponds to the number of threshold exceedances of interest over which the maxima is taken.
\(z\) is the corresponding expected value of this block maxima.
Note that the actual parametrization is in terms of excess expected mean, meaning expected mean minus threshold.
vector of length 2 containing \(z\) and \(\xi\), respectively the mean excess of the maxima of N exceedances above the threshold and the shape parameter.
sample vector
block size for threshold exceedances.
numerical tolerance for the exponential model
vector calculated by gpdN.Vfun
gpdN.ll(par, dat, N, tol=1e-5)
gpdN.score(par, dat, N)
gpdN.infomat(par, dat, N, method = c('obs', 'exp'), nobs = length(dat))
gpdN.Vfun(par, dat, N)
gpdN.phi(par, dat, N, V)
gpdN.dphi(par, dat, N, V)
gpdN.ll: log likelihood
gpdN.score: score vector
gpdN.infomat: observed information matrix for GP parametrized in terms of mean of the maximum of N exceedances and shape
gpdN.Vfun: vector implementing conditioning on approximate ancillary statistics for the TEM
gpdN.phi: canonical parameter in the local exponential family approximation
gpdN.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.