fbetagpdcon function

MLE Fitting of beta Bulk and GPD Tail Extreme Value Mixture Model with Single Continuity Constraint

MLE Fitting of beta Bulk and GPD Tail Extreme Value Mixture Model with Single Continuity Constraint

Maximum likelihood estimation for fitting the extreme value mixture model with beta 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.

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

lbetagpdcon(x, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1,
  bshape2), xi = 0, phiu = TRUE, log = TRUE)

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

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

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

Arguments

  • x: vector of sample data
  • phiu: probability of being above threshold (0,1)(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
  • bshape1: scalar beta shape 1 (positive)
  • bshape2: scalar beta shape 2 (positive)
  • u: scalar threshold over (0,1)(0, 1)
  • xi: scalar shape parameter
  • log: logical, if TRUE then log-likelihood rather than likelihood is output

Returns

Log-likelihood is given by lbetagpdcon and it's wrappers for negative log-likelihood from nlbetagpdcon

and nlubetagpdcon. Profile likelihood for single threshold given by proflubetagpdcon. Fitting function fbetagpdcon 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
bshape1 :MLE of beta shape1
bshape2 :MLE of beta shape2
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

Details

The extreme value mixture model with beta 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 dbetagpdcon for details, type help betagpdcon. Therefore, sigmau should not be included in the parameter vector if initial values are provided, making the full parameter vector (bshape1, bshape2, u, xi) if threshold is also estimated and (bshape1, bshape2, xi) for profile likelihood or fixed threshold approach.

Negative data are ignored. Values above 1 must come from GPD component, as threshold u<1.

Note

When pvector=NULL then the initial values are:

  • method of moments estimator of beta parameters assuming entire population is beta; and
  • threshold 90% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);
  • MLE of GPD shape parameter above threshold.

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd. Based on code by Anna MacDonald produced for MATLAB.

Examples

## Not run:

set.seed(1)
par(mfrow = c(2, 1))

x = rbeta(1000, shape1 = 2, shape2 = 4)
xx = seq(-0.1, 2, 0.01)
y = dbeta(xx, shape1 = 2, shape2 = 4)

# Continuity constraint
fit = fbetagpdcon(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 2))
lines(xx, y)
with(fit, lines(xx, dbetagpdcon(xx, bshape1, bshape2, u, xi), col="red"))
abline(v = fit$u, col = "red")
  
# No continuity constraint
fit2 = fbetagpd(x, phiu = FALSE)
with(fit2, lines(xx, dbetagpd(xx, bshape1, bshape2, 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 = fbetagpdcon(x, useq = seq(0.3, 0.7, length = 20))
fitfix = fbetagpdcon(x, useq = seq(0.3, 0.7, length = 20), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 2))
lines(xx, y)
with(fit, lines(xx, dbetagpdcon(xx, bshape1, bshape2, u, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dbetagpdcon(xx, bshape1, bshape2, u, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dbetagpdcon(xx, bshape1, bshape2, 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)

References

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

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

MacDonald, A. (2012). Extreme value mixture modelling with medical and industrial applications. PhD thesis, University of Canterbury, New Zealand. http://ir.canterbury.ac.nz/bitstream/10092/6679/1/thesis_fulltext.pdf

See Also

dbeta, fgpd and gpd

Other betagpd: betagpdcon, betagpd, fbetagpd

Other betagpdcon: betagpdcon, betagpd, fbetagpd

Other fbetagpdcon: betagpdcon

Author(s)

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

evmix packageRead PDF manual