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

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 between thresholds and conditional GPDs beyond thresholds. With options for profile likelihood estimation for both thresholds and fixed threshold approach.

Usage

fgkg(x, phiul = TRUE, phiur = TRUE, ulseq = NULL, urseq = 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,
  ...)

lgkg(x, lambda = NULL, ul = 0, sigmaul = 1, xil = 0,
  phiul = TRUE, ur = 0, sigmaur = 1, xir = 0, phiur = TRUE,
  bw = NULL, kernel = "gaussian", log = TRUE)

nlgkg(pvector, x, phiul = TRUE, phiur = TRUE, kernel = "gaussian",
  finitelik = FALSE)

proflugkg(ulr, pvector, x, phiul = TRUE, phiur = TRUE,
  kernel = "gaussian", method = "BFGS", control = list(maxit =
  10000), finitelik = TRUE, ...)

nlugkg(pvector, ul, ur, x, phiul = TRUE, phiur = TRUE,
  kernel = "gaussian", finitelik = FALSE)

Arguments

x	vector of sample data
phiul	probability of being below lower threshold (0, 1) or logical, see Details in help for fgng
phiur	probability of being above upper threshold (0, 1) or logical, see Details in help for fgng
ulseq	vector of lower thresholds (or scalar) to be considered in profile likelihood or NULL for no profile likelihood
urseq	vector of upper 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 ulseq/urseq, or estimated from maximum of profile likelihood evaluated at sequence of thresholds in ulseq/urseq)
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)
ul	scalar lower tail threshold
sigmaul	scalar lower tail GPD scale parameter (positive)
xil	scalar lower tail GPD shape parameter
ur	scalar upper tail threshold
sigmaur	scalar upper tail GPD scale parameter (positive)
xir	scalar upper tail GPD shape parameter
bw	scalar bandwidth for kernel (as standard deviations of kernel)
log	logical, if TRUE then log-likelihood rather than likelihood is output
ulr	vector of length 2 giving lower and upper tail thresholds or NULL for default values

Details

The extreme value mixture model with kernel density estimate for bulk and GPD for both tails 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 and fgkg for details, type help fnormgpd and help fgkg. Only the different features are outlined below for brevity.

The full parameter vector is (lambda, ul, sigmaul, xil, ur, sigmaur, xir) if thresholds are also estimated and (lambda, sigmaul, xil, sigmaur, xir) for profile likelihood or fixed threshold approach.

Cross-validation likelihood is used for KDE, but standard likelihood is used for GPD components. 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.

The tail fractions phiul and phiur are treated separately to the other parameters, to allow for all their representations. In the fitting functions fgkg and proflugkg they are logical:

• default values phiul=TRUE and phiur=TRUE - tail fractions specified by KDE distribution and survivior functions respectively and standard error is output as NA.

• phiul=FALSE and phiur=FALSE - treated as extra parameters estimated using the MLE which is the sample proportion beyond the thresholds and standard error is output.

In the likelihood functions lgkg, nlgkg and nlugkg it can be logical or numeric:

• logical - same as for fitting functions with default values phiul=TRUE and phiur=TRUE.

• numeric - any value over range (0, 1). Notice that the tail fraction probability cannot be 0 or 1 otherwise there would be no contribution from either tail or bulk components respectively. Also, phiul+phiur<1 as bulk must contribute.

If the profile likelihood approach is used, then a grid search over all combinations of both thresholds is carried out. The combinations which lead to less than 5 in any datapoints beyond the thresholds are not considered.

Value

Log-likelihood is given by lgkg and it's wrappers for negative log-likelihood from nlgkg and nlugkg. Profile likelihood for both thresholds given by proflugkg. Fitting function fgkg returns a simple list with the following elements

call:	optim call
x:	data vector x
init:	pvector
fixedu:	fixed thresholds, logical
ulseq:	lower threshold vector for profile likelihood or scalar for fixed threshold
urseq:	upper threshold vector for profile likelihood or scalar for fixed threshold
nllhuseq:	profile negative log-likelihood at each threshold pair in (ulseq, urseq)
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)
ul:	lower threshold (fixed or MLE)
sigmaul:	MLE of lower tail GPD scale
xil:	MLE of lower tail GPD shape
phiul:	MLE of lower tail fraction (bulk model or parameterised approach)
se.phiul:	standard error of MLE of lower tail fraction
ur:	upper threshold (fixed or MLE)
sigmaur:	MLE of upper tail GPD scale
xir:	MLE of upper tail GPD shape
phiur:	MLE of upper tail fraction (bulk model or parameterised approach)
se.phiur:	standard error of MLE of upper 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.

• lower threshold 10% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);

• upper threshold 90% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);

• MLE of GPD parameters beyond thresholds.

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, fkdengpdcon, fkdengpd, fkden, kdengpdcon, kdengpd, kden

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

Other gkg: fgkgcon, fkdengpd, gkgcon, gkg, kdengpd, kden

Other gkgcon: fgkgcon, fkdengpdcon, gkgcon, gkg, kdengpdcon

Other fgkg: gkg

Examples

## 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 = fgkg(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="red"))
abline(v = c(fit$ul, fit$ur), col = "red")
  
# Parameterised tail fraction
fit2 = fgkg(x, phiul = FALSE, phiur = FALSE)
with(fit2, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="blue"))
abline(v = c(fit2$ul, fit2$ur), 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 = fgkg(x, ulseq = seq(-2, -0.2, length = 10), 
 urseq = seq(0.2, 2, length = 10))
fitfix = fgkg(x, ulseq = seq(-2, -0.2, length = 10), 
 urseq = seq(0.2, 2, length = 10), fixedu = TRUE)

hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="red"))
abline(v = c(fit$ul, fit$ur), col = "red")
with(fitu, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="purple"))
abline(v = c(fitu$ul, fitu$ur), col = "purple")
with(fitfix, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="darkgreen"))
abline(v = c(fitfix$ul, fitfix$ur), 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.