fitmnormgpd: MLE Fitting of Normal Bulk and GPD Tail Interval Transition...

In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation

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.

Usage

fitmnormgpd(x, eseq = NULL, useq = NULL, fixedeu = FALSE,
  pvector = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)

litmnormgpd(x, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
  nmean, nsd), sigmau = nsd, xi = 0, log = TRUE)

nlitmnormgpd(pvector, x, finitelik = FALSE)

profleuitmnormgpd(eu, pvector, x, method = "BFGS", control = list(maxit
  = 10000), finitelik = TRUE, ...)

nleuitmnormgpd(pvector, epsilon, u, 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
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);

• MLE of GPD parameters above threshold.

Author(s)

Alfadino Akbar and Carl Scarrott [e-mail]

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

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

See Also

fnormgpd, dnorm, fgpd and gpd

Other normgpd: fgng, fhpd, flognormgpd, fnormgpdcon, fnormgpd, gngcon, gng, hpdcon, hpd, itmnormgpd, lognormgpdcon, lognormgpd, normgpdcon, normgpd

Other itmnormgpd: fitmgng, itmgng, itmnormgpd

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
fit = fitmnormgpd(x)
hist(x, breaks = seq(-6, 6, 0.1), freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, ditmnormgpd(xx, nmean, nsd, epsilon, u, sigmau, xi), col="red"))
abline(v = fit$u + fit$epsilon * seq(-1, 1), col = "red")
  
# Profile likelihood for threshold which is then fixed
fitfix = fitmnormgpd(x, eseq = seq(0, 2, 0.1), useq = seq(0, 2.5, 0.1), fixedeu = TRUE)
with(fitfix, lines(xx, ditmnormgpd(xx, nmean, nsd, epsilon, u, sigmau, xi), col="blue"))
abline(v = fitfix$u + fitfix$epsilon * seq(-1, 1), col = "blue")
legend("topright", c("True Density", "normal-GPD ITM", "Profile likelihood"),
  col=c("black", "red", "blue"), lty = 1)

## End(Not run)
  

evmix documentation built on Sept. 3, 2019, 5:07 p.m.