| fgngcon {evmix} | R Documentation |
MLE Fitting of Normal Bulk and GPD for Both Tails with Single Continuity Constraint at Both Thresholds Extreme Value Mixture Model
Description
Maximum likelihood estimation for fitting the extreme value mixture model with normal for bulk distribution between thresholds and conditional GPDs for both tails with continuity at thresholds. With options for profile likelihood estimation for both thresholds and fixed threshold approach.
Usage
fgngcon(x, phiul = TRUE, phiur = TRUE, ulseq = NULL, urseq = NULL,
fixedu = FALSE, pvector = NULL, std.err = TRUE, method = "BFGS",
control = list(maxit = 10000), finitelik = TRUE, ...)
lgngcon(x, nmean = 0, nsd = 1, ul = 0, xil = 0, phiul = TRUE,
ur = 0, xir = 0, phiur = TRUE, log = TRUE)
nlgngcon(pvector, x, phiul = TRUE, phiur = TRUE, finitelik = FALSE)
proflugngcon(ulr, pvector, x, phiul = TRUE, phiur = TRUE,
method = "BFGS", control = list(maxit = 10000), finitelik = TRUE,
...)
nlugngcon(pvector, ul, ur, x, phiul = TRUE, phiur = TRUE,
finitelik = FALSE)
Arguments
x |
vector of sample data |
phiul |
probability of being below lower threshold |
phiur |
probability of being above upper threshold |
ulseq |
vector of lower thresholds (or scalar) to be considered in profile likelihood or
|
urseq |
vector of upper thresholds (or scalar) to be considered in profile likelihood or
|
fixedu |
logical, should threshold be fixed (at either scalar value in |
pvector |
vector of initial values of parameters or |
std.err |
logical, should standard errors be calculated |
method |
optimisation method (see |
control |
optimisation control list (see |
finitelik |
logical, should log-likelihood return finite value for invalid parameters |
... |
optional inputs passed to |
nmean |
scalar normal mean |
nsd |
scalar normal standard deviation (positive) |
ul |
scalar lower tail threshold |
xil |
scalar lower tail GPD shape parameter |
ur |
scalar upper tail threshold |
xir |
scalar upper tail GPD shape parameter |
log |
logical, if |
ulr |
vector of length 2 giving lower and upper tail thresholds or
|
Details
The extreme value mixture model with normal bulk and GPD for both tails with continuity at thresholds 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 and
fgngfor details, type help fnormgpd and help fgng.
Only the different features are outlined below for brevity.
The GPD sigmaul and sigmaur parameters are now specified as function of
other parameters, see
help for dgngcon for details, type help gngcon.
Therefore, sigmaul and sigmaur should not be included in the parameter
vector if initial values are provided, making the full parameter vector
The full parameter vector is
(nmean, nsd, ul, xil, ur, xir)
if thresholds are also estimated and
(nmean, nsd, xil, xir)
for profile likelihood or fixed threshold approach.
If the profile likelihood approach is used, then a grid search over all combinations of both thresholds is carried out. The combinations which lead to less than 5 in any datapoints beyond the thresholds are not considered.
Value
Log-likelihood is given by lgngcon and it's
wrappers for negative log-likelihood from nlgngcon
and nlugngcon. Profile likelihood for both
thresholds given by proflugngcon. Fitting function
fgngcon returns a simple list with the
following elements
call: | optim call |
x: | data vector x |
init: | pvector |
fixedu: | fixed thresholds, 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 |
nllhuseq: | profile negative log-likelihood at each threshold pair in (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 |
rate: | phiu to be consistent with evd |
nllh: | minimum negative log-likelihood |
n: | total sample size |
nmean: | MLE of normal mean |
nsd: | MLE of normal standard deviation |
ul: | lower threshold (fixed or MLE) |
sigmaul: | MLE of lower tail GPD scale (estimated from other parameters) |
xil: | MLE of lower tail GPD shape |
phiul: | MLE of lower tail fraction (bulk model or parameterised approach) |
se.phiul: | standard error of MLE of lower tail fraction |
ur: | upper threshold (fixed or MLE) |
sigmaur: | MLE of upper tail GPD scale (estimated from other parameters) |
xir: | MLE of upper tail GPD shape |
phiur: | MLE of upper tail fraction (bulk model or parameterised approach) |
se.phiur: | standard error of MLE of upper tail fraction |
Acknowledgments
See Acknowledgments in
fnormgpd, type help fnormgpd. Based on code
by Xin Zhao produced for MATLAB.
Note
When pvector=NULL then the initial values are:
MLE of normal parameters assuming entire population is normal; and
lower threshold 10% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);
upper threshold 90% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);
MLE of GPD shape parameters beyond threshold.
Author(s)
Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz
References
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
Hu, Y. (2013). Extreme value mixture modelling: An R package and simulation study. MSc (Hons) thesis, University of Canterbury, New Zealand. http://ir.canterbury.ac.nz/simple-search?query=extreme&submit=Go
Zhao, X., Scarrott, C.J. Reale, M. and Oxley, L. (2010). Extreme value modelling for forecasting the market crisis. Applied Financial Econometrics 20(1), 63-72.
Mendes, B. and H. F. Lopes (2004). Data driven estimates for mixtures. Computational Statistics and Data Analysis 47(3), 583-598.
See Also
Other normgpdcon: fhpdcon,
flognormgpdcon, fnormgpdcon,
fnormgpd, gngcon,
gng, hpdcon,
hpd, normgpdcon,
normgpd
Other gng: fgng, fitmgng,
fnormgpd, gngcon,
gng, itmgng,
normgpd
Other gngcon: fgng,
fnormgpdcon, gngcon,
gng, normgpdcon
Other fgngcon: gngcon
Examples
## Not run:
set.seed(1)
par(mfrow = c(2, 1))
x = rnorm(1000)
xx = seq(-4, 4, 0.01)
y = dnorm(xx)
# Continuity constraint
fit = fgngcon(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dgngcon(xx, nmean, nsd, ul, xil, phiul,
ur, xir, phiur), col="red"))
abline(v = c(fit$ul, fit$ur), col = "red")
# No continuity constraint
fit2 = fgng(x)
with(fit2, lines(xx, dgng(xx, nmean, nsd, ul, sigmaul, xil, phiul,
ur, sigmaur, xir, phiur), col="blue"))
abline(v = c(fit2$ul, fit2$ur), col = "blue")
legend("topleft", c("True Density","No continuity constraint","With continuty constraint"),
col=c("black", "blue", "red"), lty = 1)
# Profile likelihood for initial value of threshold and fixed threshold approach
fitu = fgngcon(x, ulseq = seq(-2, -0.2, length = 10),
urseq = seq(0.2, 2, length = 10))
fitfix = fgngcon(x, ulseq = seq(-2, -0.2, length = 10),
urseq = seq(0.2, 2, length = 10), fixedu = TRUE)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dgngcon(xx, nmean, nsd, ul, xil, phiul,
ur, xir, phiur), col="red"))
abline(v = c(fit$ul, fit$ur), col = "red")
with(fitu, lines(xx, dgngcon(xx, nmean, nsd, ul, xil, phiul,
ur, xir, phiur), col="purple"))
abline(v = c(fitu$ul, fitu$ur), col = "purple")
with(fitfix, lines(xx, dgngcon(xx, nmean, nsd, ul, xil, phiul,
ur, xir, phiur), col="darkgreen"))
abline(v = c(fitfix$ul, fitfix$ur), col = "darkgreen")
legend("topright", c("True Density","Default initial value (90% quantile)",
"Prof. lik. for initial value", "Prof. lik. for fixed threshold"),
col=c("black", "red", "purple", "darkgreen"), lty = 1)
## End(Not run)