R: MLE Fitting of Normal Bulk and GPD Tail Interval Transition...
fitmnormgpd
R Documentation
MLE Fitting of Normal Bulk and GPD Tail Interval Transition Mixture Model
Description
Maximum likelihood estimation for fitting the extreme value
mixture model with the normal bulk and GPD tail interval transition mixture model.
With options for profile likelihood estimation for threshold and interval half-width,
which can both be fixed.
vector of epsilons (or scalar) to be considered in profile likelihood or
NULL for no profile likelihood
useq
vector of thresholds (or scalar) to be considered in profile likelihood or
NULL for no profile likelihood
fixedeu
logical, should threshold and epsilon be fixed
(at either scalar value in useq and eseq,
or estimated from maximum of profile likelihood evaluated at
grid of thresholds and epsilons in useq and eseq)
pvector
vector of initial values of parameters or NULL for default
values, see below
std.err
logical, should standard errors be calculated
method
optimisation method (see optim)
control
optimisation control list (see optim)
finitelik
logical, should log-likelihood return finite value for invalid parameters
...
optional inputs passed to optim
nmean
scalar normal mean
nsd
scalar normal standard deviation (positive)
epsilon
interval half-width
u
scalar threshold value
sigmau
scalar scale parameter (positive)
xi
scalar shape parameter
log
logical, if TRUE then log-likelihood rather than likelihood is output
eu
vector of epsilon and threshold pair considered in profile likelihood
Details
The extreme value mixture model with the normal bulk and GPD tail with interval
transition is fitted to the entire dataset using maximum likelihood estimation.
The estimated parameters, variance-covariance matrix and their standard errors are automatically
output.
See ditmnormgpd for explanation of normal-GPD interval
transition model, including mixing functions.
See also help for fnormgpd for mixture model fitting details.
Only the different features are outlined below for brevity.
The full parameter vector is
(nmean, nsd, epsilon, u, sigmau, xi)
if threshold and interval half-width are both estimated and
(nmean, nsd, sigmau, xi)
for profile likelihood or fixed threshold and epsilon approach.
If the profile likelihood approach is used, then it is applied to both the threshold and
epsilon parameters together. A grid search over all combinations of epsilons and thresholds
are considered. The combinations which lead to less than 5 on either side of the interval are
not considered.
A fixed threshold and epsilon approach is acheived by setting a single scalar value to each in
useq and eseq respectively.
If the profile likelihood approach is used, then a grid search over all combinations of epsilon and threshold
are carried out. The combinations which lead to less than 5 in any any interval are not considered.
Value
Log-likelihood is given by litmnormgpd and it's
wrappers for negative log-likelihood from nlitmnormgpd
and nluitmnormgpd. Profile likelihood for
threshold and interval half-width given by profluitmnormgpd.
Fitting function fitmnormgpd returns a simple list
with the following elements
call:
optim call
x:
data vector x
init:
pvector
fixedeu:
fixed epsilon and threshold, logical
useq:
threshold vector for profile likelihood or scalar for fixed threshold
eseq:
epsilon vector for profile likelihood or scalar for fixed epsilon
nllheuseq:
profile negative log-likelihood at each combination in (eseq, useq)
optim:
complete optim output
mle:
vector of MLE of parameters
cov:
variance-covariance matrix of MLE of parameters
se:
vector of standard errors of MLE of parameters
nllh:
minimum negative log-likelihood
n:
total sample size
nmean:
MLE of normal shape
nsd:
MLE of normal scale
epsilon:
MLE of transition half-width
u:
threshold (fixed or MLE)
sigmau:
MLE of GPD scale
xi:
MLE of GPD shape
Acknowledgments
See Acknowledgments in
fnormgpd, type help fnormgpd.
Note
When pvector=NULL then the initial values are:
MLE of normal parameters assuming entire population is normal; and
epsilon is MLE of normal standard deviation;
threshold 90% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);