fhpdcon: MLE Fitting of Hybrid Pareto Extreme Value Mixture Model with Single Continuity Constraint

Description

Maximum likelihood estimation for fitting the Hybrid Pareto extreme value mixture model, with only continuity at threshold and not necessarily continuous in first derivative. With options for profile likelihood estimation for threshold and fixed threshold approach.

Usage

fhpdcon(x, useq = NULL, fixedu = FALSE, pvector = NULL,
  std.err = TRUE, method = "BFGS", control = list(maxit = 10000),
  finitelik = TRUE, ...)
lhpdcon(x, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd), xi = 0,
  log = TRUE)
nlhpdcon(pvector, x, finitelik = FALSE)
profluhpdcon(u, pvector, x, method = "BFGS", control = list(maxit =
  10000), finitelik = TRUE, ...)
nluhpdcon(pvector, u, x, finitelik = FALSE)

Arguments

vector of sample data

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

lhpdcon, nlhpdcon, and nluhpdcon give the log-likelihood, negative log-likelihood and profile likelihood for threshold. Profile likelihood for single threshold is given by profluhpdcon. fhpdcon 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 (implied by `1/(1+pnorm(u,nmean,nsd))`)

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd.

Details

The hybrid Pareto model is fitted to the entire dataset using maximum likelihood estimation, with only continuity at threshold and not necessarily continuous in first derivative. The estimated parameters, variance-covariance matrix and their standard errors are automatically output.

Note that the key difference between this model (hpdcon) and the normal with GPD tail and continuity at threshold (normgpdcon) is that the latter includes the rescaling of the conditional GPD component by the tail fraction to make it an unconditional tail model. However, for the hybrid Pareto with single continuity constraint use the GPD in it's conditional form with no differential scaling compared to the bulk model.

See help for fnormgpd for details, type help fnormgpd. Only the different features are outlined below for brevity.

The profile likelihood and fixed threshold approach functionality are implemented for this version of the hybrid Pareto as it includes the threshold as a parameter. Whereas the usual hybrid Pareto does not naturally have a threshold parameter.

The GPD sigmau parameter is now specified as function of other parameters, see help for dhpdcon for details, type help hpdcon. 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

Carreau, J. and Y. Bengio (2008). A hybrid Pareto model for asymmetric fat-tailed data: the univariate case. Extremes 12 (1), 53-76.

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)

# Hybrid Pareto provides reasonable fit for some asymmetric heavy upper tailed distributions
# but not for cases such as the normal distribution

# Continuity constraint
fit = fhpdcon(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dhpdcon(xx, nmean, nsd, u, xi), col="red"))
abline(v = fit$u, col = "red")
  
# No continuity constraint
fit2 = fhpd(x)
with(fit2, lines(xx, dhpd(xx, nmean, nsd, 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 = fhpdcon(x, useq = seq(-2, 2, length = 20))
fitfix = fhpdcon(x, useq = seq(-2, 2, length = 20), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dhpdcon(xx, nmean, nsd, u, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dhpdcon(xx, nmean, nsd, u, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dhpdcon(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)
  
# Notice that if tail fraction is included a better fit is obtained
fittailfrac = fnormgpdcon(x)

par(mfrow = c(1, 1))
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dhpdcon(xx, nmean, nsd, u, xi), col="red"))
abline(v = fit$u, col = "red")
with(fittailfrac, lines(xx, dnormgpdcon(xx, nmean, nsd, u, xi), col="blue"))
abline(v = fittailfrac$u)
legend("topright", c("Standard Normal", "Hybrid Pareto Continuous", "Normal+GPD Continuous"),
  col=c("black", "red", "blue"), lty = 1)
# }
# NOT RUN {
# }

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