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.
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
)
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.
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
.
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 \(i\).
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')
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