fnormgpdcon: MLE Fitting of Normal Bulk and GPD Tail Extreme Value Mixture Model with Single Continuity Constraint

Description

Maximum likelihood estimation for fitting the extreme value mixture model with normal 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

fnormgpdcon(x, phiu = TRUE, useq = NULL, fixedu = FALSE,
  pvector = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)
lnormgpdcon(x, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  xi = 0, phiu = TRUE, log = TRUE)
nlnormgpdcon(pvector, x, phiu = TRUE, finitelik = FALSE)
proflunormgpdcon(u, pvector, x, phiu = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)
nlunormgpdcon(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

nmean

scalar normal mean

nsd

scalar normal standard deviation (positive)

scalar threshold value

scalar shape parameter

log

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

Value

Log-likelihood is given by lnormgpdcon and it's wrappers for negative log-likelihood from nlnormgpdcon and nlunormgpdcon. Profile likelihood for single threshold given by proflunormgpdcon. Fitting function fnormgpdcon 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
`nmean`:	MLE of normal mean
`nsd`:	MLE of normal standard deviation
`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.

Details

The extreme value mixture model with normal 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 full 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 dnormgpdcon for details, type help normgpdcon. Therefore, sigmau should not be included in the parameter vector if initial values are provided, making the full parameter vector (nmean, nsd, u, xi) if threshold is also estimated and (nmean, nsd, xi) for profile likelihood or fixed threshold approach.

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

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 = rnorm(1000)
xx = seq(-4, 4, 0.01)
y = dnorm(xx)

# Continuity constraint
fit = fnormgpdcon(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dnormgpdcon(xx, nmean, nsd, u, xi), col="red"))
abline(v = fit$u, col = "red")
  
# No continuity constraint
fit2 = fnormgpd(x)
with(fit2, lines(xx, dnormgpd(xx, nmean, nsd, u, sigmau, xi), col="blue"))
abline(v = fit2$u, col = "blue")
legend("topleft", 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 = fnormgpdcon(x, useq = seq(0, 3, length = 20))
fitfix = fnormgpdcon(x, useq = seq(0, 3, length = 20), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dnormgpdcon(xx, nmean, nsd, u, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dnormgpdcon(xx, nmean, nsd, u, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dnormgpdcon(xx, nmean, nsd, u, xi), col="darkgreen"))
abline(v = fitfix$u, col = "darkgreen")
legend("topleft", 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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