fkdengpd: MLE Fitting of Kernel Density Estimate for Bulk and GPD Tail Extreme Value Mixture Model

Description

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

Usage

fkdengpd(x, phiu = TRUE, useq = NULL, fixedu = FALSE,
  pvector = NULL, kernel = "gaussian", add.jitter = FALSE,
  factor = 0.1, amount = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)
lkdengpd(x, lambda = NULL, u = 0, sigmau = 1, xi = 0,
  phiu = TRUE, bw = NULL, kernel = "gaussian", log = TRUE)
nlkdengpd(pvector, x, phiu = TRUE, kernel = "gaussian",
  finitelik = FALSE)
proflukdengpd(u, pvector, x, phiu = TRUE, kernel = "gaussian",
  method = "BFGS", control = list(maxit = 10000), finitelik = TRUE,
  ...)
nlukdengpd(pvector, u, x, phiu = TRUE, kernel = "gaussian",
  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

kernel

kernel name (default = "gaussian")

add.jitter

logical, whether jitter is needed for rounded kernel centres

factor

see jitter

amount

see jitter

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

lambda

scalar bandwidth for kernel (as half-width of kernel)

scalar threshold value

sigmau

scalar scale parameter (positive)

scalar shape parameter

scalar bandwidth for kernel (as standard deviations of kernel)

log

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

Value

Log-likelihood is given by lkdengpd and it's wrappers for negative log-likelihood from nlkdengpd and nlukdengpd. Profile likelihood for single threshold given by proflukdengpd. Fitting function fkdengpd 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
`lambda`:	MLE of lambda (kernel half-width)
`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
`bw`:	MLE of bw (kernel standard deviations)
`kernel`:	kernel name

Warning

See important warnings about cross-validation likelihood estimation in fkden, type help fkden.

Acknowledgments

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

Details

The extreme value mixture model with kernel density estimate for 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 (lambda, u, sigmau, xi) if threshold is also estimated and (lambda, sigmau, xi) for profile likelihood or fixed threshold approach.

Cross-validation likelihood is used for KDE, but standard likelihood is used for GPD component. See help for fkden for details, type help fkden.

The alternate bandwidth definitions are discussed in the kernels, with the lambda as the default used in the likelihood fitting. The bw specification is the same as used in the density function.

The possible kernels are also defined in kernels with the "gaussian" as the default choice.

References

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

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

http://en.wikipedia.org/wiki/Cross-validation_(statistics)

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

Bowman, A.W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71(2), 353-360.

Duin, R.P.W. (1976). On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Transactions on Computers C25(11), 1175-1179.

MacDonald, A., Scarrott, C.J., Lee, D., Darlow, B., Reale, M. and Russell, G. (2011). A flexible extreme value mixture model. Computational Statistics and Data Analysis 55(6), 2137-2157.

Wand, M. and Jones, M.C. (1995). Kernel Smoothing. Chapman && Hall.

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)

# Bulk model based tail fraction
fit = fkdengpd(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dkdengpd(xx, x, lambda, u, sigmau, xi), col="red"))
abline(v = fit$u, col = "red")
  
# Parameterised tail fraction
fit2 = fkdengpd(x, phiu = FALSE)
with(fit2, lines(xx, dkdengpd(xx, x, lambda, 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 = fkdengpd(x, useq = seq(0, 2, length = 20))
fitfix = fkdengpd(x, useq = seq(0, 2, length = 20), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dkdengpd(xx, x, lambda, u, sigmau, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dkdengpd(xx, x, lambda, u, sigmau, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dkdengpd(xx, x, lambda, 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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