fbckdengpd() R function from [evmix]

MLE Fitting of Boundary Corrected Kernel Density Estimate for Bulk and GPD Tail Extreme Value Mixture Model

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


fbckdengpd(x, phiu = TRUE, useq = NULL, fixedu = FALSE,
  pvector = NULL, kernel = "gaussian", bcmethod = "simple",
  proper = TRUE, nn = "jf96", offset = NULL, xmax = NULL,
  add.jitter = FALSE, factor = 0.1, amount = NULL, std.err = TRUE,
  method = "BFGS", control = list(maxit = 10000), finitelik = TRUE,
  ...)

lbckdengpd(x, lambda = NULL, u = 0, sigmau = 1, xi = 0,
  phiu = TRUE, bw = NULL, kernel = "gaussian", bcmethod = "simple",
  proper = TRUE, nn = "jf96", offset = NULL, xmax = NULL,
  log = TRUE)

nlbckdengpd(pvector, x, phiu = TRUE, kernel = "gaussian",
  bcmethod = "simple", proper = TRUE, nn = "jf96", offset = NULL,
  xmax = NULL, finitelik = FALSE)

proflubckdengpd(u, pvector, x, phiu = TRUE, kernel = "gaussian",
  bcmethod = "simple", proper = TRUE, nn = "jf96", offset = NULL,
  xmax = NULL, method = "BFGS", control = list(maxit = 10000),
  finitelik = TRUE, ...)

nlubckdengpd(pvector, u, x, phiu = TRUE, kernel = "gaussian",
  bcmethod = "simple", proper = TRUE, nn = "jf96", offset = NULL,
  xmax = NULL, 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")
bcmethod: boundary correction method
proper: logical, whether density is renormalised to integrate to unity (where needed)
nn: non-negativity correction method (simple boundary correction only)
offset: offset added to kernel centres (logtrans only) or NULL
xmax: upper bound on support (copula and beta kernels only) or NULL
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: bandwidth for kernel (as half-width of kernel) or NULL
u: scalar threshold value
sigmau: scalar scale parameter (positive)
xi: scalar shape parameter
bw: bandwidth for kernel (as standard deviations of kernel) or NULL
log: logical, if TRUE then log-likelihood rather than likelihood is output

Returns

lbckdengpd, nlbckdengpd, and nlubckdengpd give the log-likelihood, negative log-likelihood and profile likelihood for threshold. Profile likelihood for single threshold is given by proflubckdengpd. fbckdengpd 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
`bcmethod` :	boundary correction method
`proper` :	logical, whether renormalisation is requested
`nn` :	non-negative correction method
`offset` :	offset for log transformation method
`xmax` :	maximum value of scaled beta or copula

Details

The extreme value mixture model with boundary corrected kernel density estimate (BCKDE) 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.

Negative data are ignored.

Cross-validation likelihood is used for BCKDE, 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.

Unlike the standard KDE, there is no general rule-of-thumb bandwidth for all these estimators, with only certain methods having a guideline in the literature, so none have been implemented. Hence, a bandwidth must always be specified.

The simple, renorm, beta1, beta2 gamma1 and gamma2

boundary corrected kernel density estimates require renormalisation, achieved by numerical integration, so are very time consuming.

Note

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

No default initial values for parameter vector are provided, so will stop evaluation if pvector is left as NULL. Avoid setting the starting value for the shape parameter to xi=0 as depending on the optimisation method it may be get stuck.

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

Boundary Correction Methods

See dbckden for details of BCKDE methods.

Warning

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

See important warnings about boundary correction approaches in dbckden, type help bckden.

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 = rgamma(500, 2, 1)
xx = seq(-0.1, 10, 0.01)
y = dgamma(xx, 2, 1)

# Bulk model based tail fraction
pinit = c(0.1, quantile(x, 0.9), 1, 0.1) # initial values required for BCKDE
fit = fbckdengpd(x, pvector = pinit, bcmethod = "cutnorm")
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 10))
lines(xx, y)
with(fit, lines(xx, dbckdengpd(xx, x, lambda, u, sigmau, xi, bcmethod = "cutnorm"), col="red"))
abline(v = fit$u, col = "red")
  
# Parameterised tail fraction
fit2 = fbckdengpd(x, phiu = FALSE, pvector = pinit, bcmethod = "cutnorm")
with(fit2, lines(xx, dbckdengpd(xx, x, lambda, u, sigmau, xi, phiu, bc = "cutnorm"), 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
pinit = c(0.1, 1, 0.1) # notice threshold dropped from initial values
fitu = fbckdengpd(x, useq = seq(1, 6, length = 20), pvector = pinit, bcmethod = "cutnorm")
fitfix = fbckdengpd(x, useq = seq(1, 6, length = 20), fixedu = TRUE, pv = pinit, bc = "cutnorm")

hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 10))
lines(xx, y)
with(fit, lines(xx, dbckdengpd(xx, x, lambda, u, sigmau, xi, bc = "cutnorm"), col="red"))
abline(v = fit$u, col = "red")
with(fitu, lines(xx, dbckdengpd(xx, x, lambda, u, sigmau, xi, bc = "cutnorm"), col="purple"))
abline(v = fitu$u, col = "purple")
with(fitfix, lines(xx, dbckdengpd(xx, x, lambda, u, sigmau, xi, bc = "cutnorm"), 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/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.

MacDonald, A., C. J. Scarrott, and D. S. Lee (2011). Boundary correction, consistency and robustness of kernel densities using extreme value theory. Submitted. Available from: http://www.math.canterbury.ac.nz/~c.scarrott.

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

Author(s)

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

evmix package Read PDF manual

Maintainer: Carl Scarrott
License: GPL-3
Last published: 2019-09-03
http://www.math.canterbury.ac.nz/~c.scarrott/evmix

fbckdengpd function