fgammagpd() R function from [evmix]

MLE Fitting of Gamma Bulk and GPD Tail Extreme Value Mixture Model

Maximum likelihood estimation for fitting the extreme value mixture model with gamma for bulk distribution upto the threshold and conditional GPD above threshold. With options for profile likelihood estimation for threshold and fixed threshold approach.


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

lgammagpd(x, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), sigmau = sqrt(gshape) * gscale, xi = 0, phiu = TRUE,
  log = TRUE)

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

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

nlugammagpd(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
sigmau: scalar scale parameter (positive)
xi: scalar shape parameter
log: logical, if TRUE then log-likelihood rather than likelihood is output

Returns

Log-likelihood is given by lgammagpd and it's wrappers for negative log-likelihood from nlgammagpd

and nlugammagpd. Profile likelihood for single threshold given by proflugammagpd. Fitting function fgammagpd 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
`xi` :	MLE of GPD shape
`phiu` :	MLE of tail fraction (bulk model or parameterised approach)
`se.phiu` :	standard error of MLE of tail fraction

Details

The extreme value mixture model with gamma bulk and GPD tail 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 full parameter vector is (gshape, gscale, u, sigmau, xi) if threshold is also estimated and (gshape, gscale, sigmau, xi) for profile likelihood or fixed threshold approach.

Non-positive data are ignored as likelihood is infinite, except for gshape=1.

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 parameters above threshold.

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd.

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)

# Bulk model based tail fraction
fit = fgammagpd(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 8))
lines(xx, y)
with(fit, lines(xx, dgammagpd(xx, gshape, gscale, u, sigmau, xi), col="red"))
abline(v = fit$u, col = "red")
  
# Parameterised tail fraction
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","Bulk Tail Fraction","Parameterised Tail Fraction"),
  col=c("black", "red", "blue"), lty = 1)
  
# Profile likelihood for initial value of threshold and fixed threshold approach
fitu = fgammagpd(x, useq = seq(1, 5, length = 20))
fitfix = fgammagpd(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, dgammagpd(xx, gshape, gscale, u, sigmau, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dgammagpd(xx, gshape, gscale, u, sigmau, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dgammagpd(xx, gshape, gscale, u, sigmau, 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)

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.

Author(s)

Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz

evmix package Read PDF manual

Maintainer: Carl Scarrott
License: GPL-3
Last published: 2019-09-03
http://www.math.canterbury.ac.nz/~c.scarrott/evmix

fgammagpd function