fgng() R function from [evmix]

MLE Fitting of Normal Bulk and GPD for Both Tails Extreme Value Mixture Model

Maximum likelihood estimation for fitting the extreme value mixture model with normal for bulk distribution between thresholds and conditional GPDs beyond thresholds. With options for profile likelihood estimation for both thresholds and fixed threshold approach.


fgng(x, phiul = TRUE, phiur = TRUE, ulseq = NULL, urseq = NULL,
  fixedu = FALSE, pvector = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)

lgng(x, nmean = 0, nsd = 1, ul = 0, sigmaul = 1, xil = 0,
  phiul = TRUE, ur = 0, sigmaur = 1, xir = 0, phiur = TRUE,
  log = TRUE)

nlgng(pvector, x, phiul = TRUE, phiur = TRUE, finitelik = FALSE)

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

nlugng(pvector, ul, ur, x, phiul = TRUE, phiur = TRUE,
  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
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
nmean: scalar normal mean
nsd: scalar normal standard deviation (positive)
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
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

Returns

Log-likelihood is given by lgng and it's wrappers for negative log-likelihood from nlgng

and nlugng. Profile likelihood for both thresholds given by proflugng. Fitting function fgng 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
`nmean` :	MLE of normal mean
`nsd` :	MLE of normal standard deviation
`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

Details

The extreme value mixture model with normal 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 for details, type help fnormgpd. Only the different features are outlined below for brevity.

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

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

default values phiul=TRUE and phiur=TRUE - tail fractions specified by normal distribution pnorm(ul, nmean, nsd) and survivior functions 1-pnorm(ur, nmean, nsd) 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 lgng, nlgng and nlugng

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.

Note

When pvector=NULL then the initial values are:

MLE of normal parameters assuming entire population is normal; and
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 threshold.

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd. Based on code by Xin Zhao produced for MATLAB.

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 = fgng(x)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
lines(xx, y)
with(fit, lines(xx, dgng(xx, nmean, nsd, ul, sigmaul, xil, phiul, 
   ur, sigmaur, xir, phiur), col="red"))
abline(v = c(fit$ul, fit$ur), col = "red")
  
# Parameterised tail fraction
fit2 = fgng(x, phiul = FALSE, phiur = FALSE)
with(fit2, lines(xx, dgng(xx, nmean, nsd, 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 = fgng(x, ulseq = seq(-2, -0.2, length = 10), 
 urseq = seq(0.2, 2, length = 10))
fitfix = fgng(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, dgng(xx, nmean, nsd, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="red"))
abline(v = c(fit$ul, fit$ur), col = "red")
with(fitu, lines(xx, dgng(xx, nmean, nsd, ul, sigmaul, xil, phiul,
   ur, sigmaur, xir, phiur), col="purple"))
abline(v = c(fitu$ul, fitu$ur), col = "purple")
with(fitfix, lines(xx, dgng(xx, nmean, nsd, 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)

References

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

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

Zhao, X., Scarrott, C.J. Reale, M. and Oxley, L. (2010). Extreme value modelling for forecasting the market crisis. Applied Financial Econometrics 20(1), 63-72.

Mendes, B. and H. F. Lopes (2004). Data driven estimates for mixtures. Computational Statistics and Data Analysis 47(3), 583-598.

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

fgng function