The method uses the pseudo-polar transformation for suitable norms, transforming
the data to pseudo-observations, than marginally to unit Frechet or unit Pareto.
Empirical or Euclidean weights are computed and returned alongside with the angular and
radial sample for values above threshold(s) th
, specified in terms of quantiles
of the radial component R
or marginal quantiles. Only complete tuples are kept.
angmeas(
x,
th,
Rnorm = c("l1", "l2", "linf"),
Anorm = c("l1", "l2", "linf", "arctan"),
marg = c("Frechet", "Pareto"),
wgt = c("Empirical", "Euclidean"),
region = c("sum", "min", "max"),
is.angle = FALSE
)
a list with arguments ang
for the \(d-1\) pseudo-angular sample, rad
with the radial component
and possibly wts
if Rnorm='l1'
and the empirical likelihood algorithm converged. The Euclidean algorithm always returns weights even if some of these are negative.
a list with components
ang
matrix of pseudo-angular observations
rad
vector of radial contributions
wts
empirical or Euclidean likelihood weights for angular observations
an n
by d
sample matrix
threshold of length 1 for 'sum'
, or d
marginal thresholds otherwise.
character string indicating the norm for the radial component.
character string indicating the norm for the angular component. arctan
is only implemented for \(d=2\)
character string indicating choice of marginal transformation, either to Frechet or Pareto scale
character string indicating weighting function for the equation. Can be based on Euclidean or empirical likelihood for the mean
character string specifying which observations to consider (and weight). 'sum'
corresponds to a radial threshold
\(\sum x_i > \)th
, 'min'
to \(\min x_i >\)th
and 'max'
to \(\max x_i >\)th
.
logical indicating whether observations are already angle with respect to region
. Default to FALSE
.
Leo Belzile
The empirical likelihood weighted mean problem is implemented for all thresholds,
while the Euclidean likelihood is only supported for diagonal thresholds specified
via region=sum
.
Einmahl, J.H.J. and J. Segers (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution, Annals of Statistics, 37(5B), 2953--2989.
de Carvalho, M. and B. Oumow and J. Segers and M. Warchol (2013). A Euclidean likelihood estimator for bivariate tail dependence, Comm. Statist. Theory Methods, 42(7), 1176--1192.
Owen, A.B. (2001). Empirical Likelihood, CRC Press, 304p.
x <- rmev(n=25, d=3, param=0.5, model='log')
wts <- angmeas(x=x, th=0, Rnorm='l1', Anorm='l1', marg='Frechet', wgt='Empirical')
wts2 <- angmeas(x=x, Rnorm='l2', Anorm='l2', marg='Pareto', th=0)
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