Maximum likelihood estimation for fitting the extreme value mixture model with normal for bulk distribution between thresholds, conditional GPDs beyond thresholds and interval transition. With options for profile likelihood estimation for both thresholds and interval half-width, which can also be fixed.
fitmgng(x, eseq = NULL, ulseq = NULL, urseq = NULL, fixedeu = FALSE,
pvector = NULL, std.err = TRUE, method = "BFGS", control = list(maxit
= 10000), finitelik = TRUE, ...)litmgng(x, nmean = 0, nsd = 1, epsilon = nsd, ul = 0, sigmaul = 1,
xil = 0, ur = 0, sigmaur = 1, xir = 0, log = TRUE)
nlitmgng(pvector, x, finitelik = FALSE)
profleuitmgng(eulr, pvector, x, method = "BFGS", control = list(maxit =
10000), finitelik = TRUE, ...)
nleuitmgng(pvector, epsilon, ul, ur, x, finitelik = FALSE)
vector of sample data
vector of epsilons (or scalar) to be considered in profile likelihood or
NULL for no profile likelihood
vector of lower thresholds (or scalar) to be considered in profile likelihood or
NULL for no profile likelihood
vector of upper thresholds (or scalar) to be considered in profile likelihood or
NULL for no profile likelihood
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)
vector of initial values of parameters or NULL for default
values, see below
logical, should standard errors be calculated
optimisation method (see optim)
optimisation control list (see optim)
logical, should log-likelihood return finite value for invalid parameters
optional inputs passed to optim
scalar normal mean
scalar normal standard deviation (positive)
interval half-width
lower tail threshold
lower tail GPD scale parameter (positive)
lower tail GPD shape parameter
upper tail threshold
upper tail GPD scale parameter (positive)
upper tail GPD shape parameter
logical, if TRUE then log-likelihood rather than likelihood is output
vector of epsilon, lower and upper thresholds considered in profile likelihood
Log-likelihood is given by litmgng and it's
wrappers for negative log-likelihood from nlitmgng
and nluitmgng. Profile likelihood for
thresholds and interval half-width given by profluitmgng.
Fitting function fitmgng returns a simple list with the
following elements
call: |
optim call |
x: |
data vector x |
init: |
pvector |
fixedeu: |
fixed epsilon and threshold, logical |
ulseq: |
lower threshold vector for profile likelihood or scalar for fixed threshold |
urseq: |
upper threshold vector for profile likelihood or scalar for fixed threshold |
eseq: |
interval half-width vector for profile likelihood or scalar for fixed threshold |
nllheuseq: |
profile negative log-likelihood at each combination in (eseq, ulseq, urseq) |
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 mean |
nsd: |
MLE of normal standard deviation |
epsilon: |
MLE of transition half-width |
ul: |
lower threshold (fixed or MLE) |
sigmaul: |
MLE of lower tail GPD scale |
xil: |
MLE of lower tail GPD shape |
ur: |
upper threshold (fixed or MLE) |
sigmaur: |
MLE of upper tail GPD scale |
xir: |
MLE of upper tail GPD shape |
See Acknowledgments in
fnormgpd, type help fnormgpd. Based on code
by Xin Zhao produced for MATLAB.
The extreme value mixture model with the normal bulk and GPD for both tails 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 ditmgng for explanation of GPD-normal-GPD interval
transition model, including mixing functions.
See also help for fnormgpd for details, type help fnormgpd.
Only the different features are outlined below for brevity.
The full parameter vector is
(nmean, nsd, epsilon, ul, sigmaul, xil,
ur, sigmaur, xir)
if thresholds and interval half-width are also estimated and
(nmean, nsd, sigmaul, xil, sigmaur, xir)
for profile likelihood or fixed threshold approach.
If the profile likelihood approach is used, then a grid search over all combinations of epsilons and both thresholds are carried out. The combinations which lead to less than 5 in any component outside of the intervals are not considered.
A fixed pair of thresholds and epsilon approach is acheived by setting a single
scalar value to each in ulseq, urseq and eseq respectively.
http://www.math.canterbury.ac.nz/~c.scarrott/evmix
http://en.wikipedia.org/wiki/Normal_distribution
http://en.wikipedia.org/wiki/Generalized_Pareto_distribution
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
Holden, L. and Haug, O. (2013). A mixture model for unsupervised tail estimation. arxiv:0902.4137
Other normgpd normgpdcon gng gngcon fnormgpd fnormgpdcon fgng fgngcon: fgngcon,
fgng, fnormgpdcon,
fnormgpd, gngcon,
gng, itmgng,
normgpdcon, normgpd
# NOT RUN {
set.seed(1)
par(mfrow = c(1, 1))
x = rnorm(1000)
xx = seq(-4, 4, 0.01)
y = dnorm(xx)
# MLE for complete parameter set (not recommended!)
fit = fitmgng(x)
hist(x, breaks = seq(-6, 6, 0.1), freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, ditmgng(xx, nmean, nsd, epsilon, ul, sigmaul, xil,
ur, sigmaur, xir), col="red"))
abline(v = fit$ul + fit$epsilon * seq(-1, 1), col = "red")
abline(v = fit$ur + fit$epsilon * seq(-1, 1), col = "darkred")
# Profile likelihood for threshold which is then fixed
fitfix = fitmgng(x, eseq = seq(0, 2, 0.1), ulseq = seq(-2.5, 0, 0.25),
urseq = seq(0, 2.5, 0.25), fixedeu = TRUE)
with(fitfix, lines(xx, ditmgng(xx, nmean, nsd, epsilon, ul, sigmaul, xil,
ur, sigmaur, xir), col="blue"))
abline(v = fitfix$ul + fitfix$epsilon * seq(-1, 1), col = "blue")
abline(v = fitfix$ur + fitfix$epsilon * seq(-1, 1), col = "darkblue")
legend("topright", c("True Density", "GPD-normal-GPD ITM", "Profile likelihood"),
col=c("black", "red", "blue"), lty = 1)
# }
# NOT RUN {
# }
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