Dirichlet mixture smoothing of the angular measure

This function computes the empirical or Euclidean likelihood estimates of the spectral measure and uses the points returned from a call to angmeas to compute the Dirichlet mixture smoothing of de Carvalho, Warchol and Segers (2012), placing a Dirichlet kernel at each observation.

Usage

angmeasdir(
  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

an invisible list with components

nu bandwidth parameter obtained by cross-validation;
dirparmat n by d matrix of Dirichlet parameters for the mixtures;
wts mixture weights.

Details

The cross-validation bandwidth is the solution of $$\max_{\nu} \sum_{i=1}^n \log \left\{ \sum_{k=1,k \neq i}^n p_{k, -i} f(\mathbf{w}_i; \nu \mathbf{w}_k)\right\},$$ where $f$ is the density of the Dirichlet distribution, $p_{k, -i}$ is the Euclidean weight obtained from estimating the Euclidean likelihood problem without observation $i$.

Examples

set.seed(123)
x <- rmev(n=100, d=2, param=0.5, model='log')
out <- angmeasdir(x=x, th=0, Rnorm='l1', Anorm='l1', marg='Frechet', wgt='Empirical')