R: MLE Fitting of Gamma Bulk and GPD Tail Extreme Value Mixture...
fgammagpd
R Documentation
MLE Fitting of Gamma Bulk and GPD Tail Extreme Value Mixture Model
Description
Maximum likelihood estimation for fitting the extreme value
mixture model with gamma for bulk distribution upto the threshold and conditional
GPD above threshold. With options for profile likelihood estimation for threshold and
fixed threshold approach.
probability of being above threshold (0, 1) or logical, see Details in
help for fnormgpd
useq
vector of thresholds (or scalar) to be considered in profile likelihood or
NULL for no profile likelihood
fixedu
logical, should threshold be fixed (at either scalar value in useq,
or estimated from maximum of profile likelihood evaluated at
sequence of thresholds in useq)
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
gshape
scalar gamma shape (positive)
gscale
scalar gamma scale (positive)
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
Details
The extreme value mixture model with gamma bulk and GPD tail is
fitted to the entire dataset using maximum likelihood estimation. The estimated
parameters, variance-covariance matrix and their standard errors are automatically
output.
See help for fnormgpd for details, type help fnormgpd.
Only the different features are outlined below for brevity.
The full parameter vector is
(gshape, gscale, u, sigmau, xi) if threshold is also estimated and
(gshape, gscale, sigmau, xi) for profile likelihood or fixed threshold approach.
Non-positive data are ignored as likelihood is infinite, except for gshape=1.
Value
Log-likelihood is given by lgammagpd and it's
wrappers for negative log-likelihood from nlgammagpd
and nlugammagpd. Profile likelihood for single
threshold given by proflugammagpd. Fitting function
fgammagpd returns a simple list with the
following elements
call:
optim call
x:
data vector x
init:
pvector
fixedu:
fixed threshold, logical
useq:
threshold vector for profile likelihood or scalar for fixed threshold
nllhuseq:
profile negative log-likelihood at each threshold in 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
rate:
phiu to be consistent with evd
nllh:
minimum negative log-likelihood
n:
total sample size
gshape:
MLE of gamma shape
gscale:
MLE of gamma scale
u:
threshold (fixed or MLE)
sigmau:
MLE of GPD scale
xi:
MLE of GPD shape
phiu:
MLE of tail fraction (bulk model or parameterised approach)
se.phiu:
standard error of MLE of tail fraction
Acknowledgments
See Acknowledgments in
fnormgpd, type help fnormgpd.
Note
When pvector=NULL then the initial values are:
approximation of MLE of gamma parameters assuming entire population is gamma; and
threshold 90% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);
Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value
threshold estimation and uncertainty quantification. REVSTAT - Statistical
Journal 10(1), 33-59. Available from http://www.ine.pt/revstat/pdf/rs120102.pdf
Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004). Bayesian analysis of extreme
events with threshold estimation. Statistical Modelling. 4(3), 227-244.