fgng: MLE Fitting of Normal Bulk and GPD for Both Tails Extreme...

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

Usage

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

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.

Value

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

Acknowledgments

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

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.

Author(s)

Yang Hu and Carl Scarrott [e-mail]

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.

See Also

dnorm, fgpd and gpd

Other normgpd: fhpd, fitmnormgpd, flognormgpd, fnormgpdcon, fnormgpd, gngcon, gng, hpdcon, hpd, itmnormgpd, lognormgpdcon, lognormgpd, normgpdcon, normgpd

Other gng: fgngcon, fitmgng, fnormgpd, gngcon, gng, itmgng, normgpd

Other gngcon: fgngcon, fnormgpdcon, gngcon, gng, normgpdcon

Other fgng: gng

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)
  

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