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Generalized Pareto distribution (return level parametrization)

Source: R/univdist.R
gpdr.Rd

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

Arguments

par

vector of length 2 containing \(y_m\) and \(\xi\), respectively the \(m\)-year return level 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 gpdr.Vfun

Details

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.

Usage

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)

Functions

  • 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

Author

Leo Belzile