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Rank-based transformation to angular measure

Source: R/specdens.R
angmeas.Rd

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.

Usage

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
)

Arguments

x

an n by d sample matrix

th

threshold of length 1 for 'sum', or d marginal thresholds otherwise.

Rnorm

character string indicating the norm for the radial component.

Anorm

character string indicating the norm for the angular component. arctan is only implemented for \(d=2\)

marg

character string indicating choice of marginal transformation, either to Frechet or Pareto scale

wgt

character string indicating weighting function for the equation. Can be based on Euclidean or empirical likelihood for the mean

region

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.

is.angle

logical indicating whether observations are already angle with respect to region. Default to FALSE.

Value

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

Details

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.

References

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.

Author

Leo Belzile

Examples

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)