fgammagpdcon: MLE Fitting of Gamma Bulk and GPD Tail Extreme Value Mixture...

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 gamma for bulk distribution upto the threshold and conditional GPD above threshold with continuity at threshold. With options for profile likelihood estimation for threshold and fixed threshold approach.

Usage

fgammagpdcon(x, phiu = TRUE, useq = NULL, fixedu = FALSE,
  pvector = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)

lgammagpdcon(x, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), xi = 0, phiu = TRUE, log = TRUE)

nlgammagpdcon(pvector, x, phiu = TRUE, finitelik = FALSE)

proflugammagpdcon(u, pvector, x, phiu = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)

nlugammagpdcon(pvector, u, x, phiu = TRUE, finitelik = FALSE)

Arguments

x	vector of sample data
phiu	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
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 with continuity at threshold 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 GPD sigmau parameter is now specified as function of other parameters, see help for dgammagpdcon for details, type help gammagpdcon. Therefore, sigmau should not be included in the parameter vector if initial values are provided, making the full parameter vector (gshape, gscale, u, xi) if threshold is also estimated and (gshape, gscale, 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 lgammagpdcon and it's wrappers for negative log-likelihood from nlgammagpdcon and nlugammagpdcon. Profile likelihood for single threshold given by proflugammagpdcon. Fitting function fgammagpdcon 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 (estimated from other parameters)
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);

• MLE of GPD shape parameter above threshold.

Author(s)

Yang Hu and Carl Scarrott [e-mail]

References

http://www.math.canterbury.ac.nz/~c.scarrott/evmix

http://en.wikipedia.org/wiki/Gamma_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

Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling. 4(3), 227-244.

See Also

dgamma, fgpd and gpd

Other gammagpd: fgammagpd, fmgammagpd, fmgamma, gammagpdcon, gammagpd, mgammagpd

Other gammagpdcon: fgammagpd, fmgammagpdcon, gammagpdcon, gammagpd, mgammagpdcon

Other mgammagpdcon: fmgammagpdcon, fmgammagpd, fmgamma, gammagpdcon, mgammagpdcon, mgammagpd, mgamma

Other fgammagpdcon: gammagpdcon

Examples

## Not run: 
set.seed(1)
par(mfrow = c(2, 1))

x = rgamma(1000, shape = 2)
xx = seq(-0.1, 8, 0.01)
y = dgamma(xx, shape = 2)

# Continuity constraint
fit = fgammagpdcon(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 8))
lines(xx, y)
with(fit, lines(xx, dgammagpdcon(xx, gshape, gscale, u, xi), col="red"))
abline(v = fit$u, col = "red")
  
# No continuity constraint
fit2 = fgammagpd(x, phiu = FALSE)
with(fit2, lines(xx, dgammagpd(xx, gshape, gscale, u, sigmau, xi, phiu), col="blue"))
abline(v = fit2$u, col = "blue")
legend("topright", 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 = fgammagpdcon(x, useq = seq(1, 5, length = 20))
fitfix = fgammagpdcon(x, useq = seq(1, 5, length = 20), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 8))
lines(xx, y)
with(fit, lines(xx, dgammagpdcon(xx, gshape, gscale, u, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dgammagpdcon(xx, gshape, gscale, u, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dgammagpdcon(xx, gshape, gscale, u, xi), col="darkgreen"))
abline(v = fitfix$u, 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)
  

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