evmix (version 2.11)

fitmgng: MLE Fitting of Normal Bulk and GPD for Both Tails Interval Transition Mixture Model

Description

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.

Usage

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)

Arguments

x

vector of sample data

eseq

vector of epsilons (or scalar) to be considered in profile likelihood or NULL for no profile likelihood

ulseq

vector of lower thresholds (or scalar) to be considered in profile likelihood or NULL for no profile likelihood

urseq

vector of upper 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

ul

lower tail threshold

sigmaul

lower tail GPD scale parameter (positive)

xil

lower tail GPD shape parameter

ur

upper tail threshold

sigmaur

upper tail GPD scale parameter (positive)

xir

upper tail GPD shape parameter

log

logical, if TRUE then log-likelihood rather than likelihood is output

eulr

vector of epsilon, lower and upper thresholds considered in profile likelihood

Value

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

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd. Based on code by Xin Zhao produced for MATLAB.

Details

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.

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

Holden, L. and Haug, O. (2013). A mixture model for unsupervised tail estimation. arxiv:0902.4137

See Also

fgng, dnorm, fgpd and gpd

Other normgpd normgpdcon gng gngcon fnormgpd fnormgpdcon fgng fgngcon: fgngcon, fgng, fnormgpdcon, fnormgpd, gngcon, gng, itmgng, normgpdcon, normgpd

Examples

# 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 {
  
# }