fweibullgpd: MLE Fitting of Weibull Bulk and GPD Tail Extreme Value Mixture Model

Description

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

Usage

fweibullgpd(x, phiu = TRUE, useq = NULL, fixedu = FALSE,
  pvector = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)
lweibullgpd(x, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
  wscale), sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale *
  gamma(1 + 1/wshape))^2), xi = 0, phiu = TRUE, log = TRUE)
nlweibullgpd(pvector, x, phiu = TRUE, finitelik = FALSE)
profluweibullgpd(u, pvector, x, phiu = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)
nluweibullgpd(pvector, u, x, phiu = TRUE, finitelik = FALSE)

Arguments

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

wshape

scalar Weibull shape (positive)

wscale

scalar Weibull scale (positive)

scalar threshold value

sigmau

scalar scale parameter (positive)

scalar shape parameter

log

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

Value

Log-likelihood is given by lweibullgpd and it's wrappers for negative log-likelihood from nlweibullgpd and nluweibullgpd. Profile likelihood for single threshold given by profluweibullgpd. Fitting function fweibullgpd 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
`wshape`:	MLE of Weibull shape
`wscale`:	MLE of Weibull 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

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd.

Details

The extreme value mixture model with Weibull 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 (wshape, wscale, u, sigmau, xi) if threshold is also estimated and (wshape, wscale, sigmau, xi) for profile likelihood or fixed threshold approach.

Non-positive data are ignored (f(0) is infinite for wshape<1).

References

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

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

Examples

Run this code

# NOT RUN {
set.seed(1)
par(mfrow = c(2, 1))

x = rweibull(1000, shape = 2)
xx = seq(-0.1, 4, 0.01)
y = dweibull(xx, shape = 2)

# Bulk model based tail fraction
fit = fweibullgpd(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 4))
lines(xx, y)
with(fit, lines(xx, dweibullgpd(xx, wshape, wscale, u, sigmau, xi), col="red"))
abline(v = fit$u, col = "red")
  
# Parameterised tail fraction
fit2 = fweibullgpd(x, phiu = FALSE)
with(fit2, lines(xx, dweibullgpd(xx, wshape, wscale, 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 = fweibullgpd(x, useq = seq(0.5, 2, length = 20))
fitfix = fweibullgpd(x, useq = seq(0.5, 2, length = 20), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 4))
lines(xx, y)
with(fit, lines(xx, dweibullgpd(xx, wshape, wscale, u, sigmau, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dweibullgpd(xx, wshape, wscale, u, sigmau, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dweibullgpd(xx, wshape, wscale, 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)
# }
# NOT RUN {
  
# }

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