lbckden(x, lambda = NULL, extracentres = NULL,
bcmethod = "simple", proper = TRUE, nn = "jf96",
offset = 0, xmax = Inf, log = TRUE)
nlbckden(lambda, x, extracentres = NULL,
bcmethod = "simple", proper = TRUE, nn = "jf96",
offset = 0, xmax = Inf, finitelik = FALSE)NULL (default)NULLfbckdenfbckden.
They are designed to be used for MLE in
fbckden but are available
for wider usage, e.g. constructing your own extreme value
mixture models.
All of the boundary correction methods available in
bckden are permitted.
See fkden and
fgpd for full details.
The cross-validation likelihood is obtained by leaving
each point out in turn, obtaining the usual KDE and
evaluate at the point left out:
$$L(\lambda)\prod_{i=1}^{n} \hat{f}_{-i}(x_i)$$ where
$$\hat{f}_{-i}(x_i) = \frac{1}{(n-1)\lambda}
\sum_{j=1: j\ne i}^{n} K(\frac{x_i - x_j}{\lambda})$$ is
the KDE obtained when the $i$th datapoint is dropped
but is evaluated at $x_i$.
Normally for likelihood estimation of the bandwidth the
kernel centres and the data where the likelihood is
evaluated are the same. However, when using KDE for
extreme value mixture modelling the likelihood only those
data in the bulk of the distribution should contribute to
the likelihood, but all the data (including those beyond
the threshold) should contribute to the density estimate.
The extracentres option allows the use to specify
extra kernel centres used in estimating the density, but
not evaluated in the likelihood. The default is to just
use the existing data, so extracentres=NULL.
Log-likelihood calculations are carried out in
lbckden, which takes
bandwidth in the same form as distribution functions. The
negative log-likelihood is a wrapper for
lbckden, designed towards
making it useable for optimisation (e.g. parameters are
given a vector as first input).
The function lbckden carries
out the calculations for the log-likelihood directly,
which can be exponentiated to give actual likelihood
using (log=FALSE).density