fkdengpdcon: MLE Fitting of Kernel Density Estimate for Bulk and GPD Tail...

In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation

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 continuity at threshold. With options for profile likelihood estimation for threshold and fixed threshold approach.

Usage

fkdengpdcon(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, ...)

lkdengpdcon(x, lambda = NULL, u = 0, xi = 0, phiu = TRUE,
  bw = NULL, kernel = "gaussian", log = TRUE)

nlkdengpdcon(pvector, x, phiu = TRUE, kernel = "gaussian",
  finitelik = FALSE)

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

nlukdengpdcon(pvector, u, x, phiu = TRUE, kernel = "gaussian",
  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
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)
u	scalar threshold value
xi	scalar shape parameter
bw	scalar bandwidth for kernel (as standard deviations of kernel)
log	logical, if TRUE then log-likelihood rather than likelihood is output

Details

The extreme value mixture model with kernel density estimate for 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 dkdengpdcon for details, type help kdengpdcon. Therefore, sigmau should not be included in the parameter vector if initial values are provided, making the full parameter vector (lambda, u, xi) if threshold is also estimated and (lambda, 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.

Value

Log-likelihood is given by lkdengpdcon and it's wrappers for negative log-likelihood from nlkdengpdcon and nlukdengpdcon. Profile likelihood for single threshold given by proflukdengpdcon. Fitting function fkdengpdcon 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 (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
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.

Note

The data and kernel centres are both vectors. Infinite and missing sample values (and kernel centres) are dropped.

When pvector=NULL then the initial values are:

• normal reference rule for bandwidth, using the bw.nrd0 function, which is consistent with the density function. At least two kernel centres must be provided as the variance needs to be estimated.

• 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 [e-mail]

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.

See Also

kernels, kfun, density, bw.nrd0 and dkde in ks package. fgpd and gpd.

Other kden: bckden, fbckden, fgkgcon, fgkg, fkdengpd, fkden, kdengpdcon, kdengpd, kden

Other kdengpd: bckdengpd, fbckdengpd, fgkg, fkdengpd, fkden, gkg, kdengpdcon, kdengpd, kden

Other kdengpdcon: bckdengpdcon, fbckdengpdcon, fgkgcon, fkdengpd, gkgcon, kdengpdcon, kdengpd

Other gkgcon: fgkgcon, fgkg, gkgcon, gkg, kdengpdcon

Other bckdengpdcon: bckdengpdcon, bckdengpd, bckden, fbckdengpdcon, fbckdengpd, fbckden, gkgcon, kdengpdcon

Other fkdengpdcon: kdengpdcon

Examples

## 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 = fkdengpdcon(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dkdengpdcon(xx, x, lambda, u, xi), col="red"))
abline(v = fit$u, col = "red")
  
# No continuity constraint
fit2 = fkdengpdcon(x)
with(fit2, lines(xx, dkdengpdcon(xx, x, lambda, u, 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 = fkdengpdcon(x, useq = seq(0, 2, length = 20))
fitfix = fkdengpdcon(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, dkdengpdcon(xx, x, lambda, u, xi), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dkdengpdcon(xx, x, lambda, u, xi), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dkdengpdcon(xx, x, lambda, 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)
  

evmix documentation built on Sept. 3, 2019, 5:07 p.m.