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

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

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)

scalar lower tail threshold

sigmaul

scalar lower tail GPD scale parameter (positive)

xil

scalar lower tail GPD shape parameter

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

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.

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.

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.

Examples

Run this code

# 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)
# }
# NOT RUN {
  
# }

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