R: MLE Fitting of beta Bulk and GPD Tail Extreme Value Mixture...

fbetagpdcon {evmix}

R Documentation

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

Description

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.

Usage

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)` 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)`
`xi`	scalar shape parameter
`log`	logical, if `TRUE` then log-likelihood rather than likelihood is output

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.

Value

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

Acknowledgments

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

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.

Author(s)

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

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

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)

[Package evmix version 2.12 Index]