fdwm {evmix}

MLE Fitting of Dynamically Weighted Mixture Model

Description

Maximum likelihood estimation for fitting the dynamically weighted mixture model

Usage

fdwm(x, pvector = NULL, std.err = TRUE, method = "BFGS",
  control = list(maxit = 10000), finitelik = TRUE, ...)

ldwm(x, wshape = 1, wscale = 1, cmu = 1, ctau = 1,
  sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 +
  1/wshape))^2), xi = 0, log = TRUE)

nldwm(pvector, x, finitelik = FALSE)

Arguments

x

vector of sample data

pvector

vector of initial values of parameters (wshape, wscale, cmu, ctau, sigmau, xi) or NULL

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

wshape

Weibull shape (positive)

wscale

Weibull scale (positive)

cmu

Cauchy location

ctau

Cauchy scale

sigmau

scalar scale parameter (positive)

xi

scalar shape parameter

log

logical, if TRUE then log-likelihood rather than likelihood is output

Details

The dynamically weighted mixture model is fitted to the entire dataset using maximum likelihood estimation. The estimated parameters, variance-covariance matrix and their standard errors are automatically output.

The log-likelihood and negative log-likelihood are also provided for wider usage, e.g. constructing profile likelihood functions. The parameter vector pvector must be specified in the negative log-likelihood nldwm.

Log-likelihood calculations are carried out in ldwm, which takes parameters as inputs in the same form as distribution functions. The negative log-likelihood is a wrapper for ldwm, designed towards making it useable for optimisation (e.g. parameters are given a vector as first input).

Non-negative data are ignored.

Missing values (NA and NaN) are assumed to be invalid data so are ignored, which is inconsistent with the evd library which assumes the missing values are below the threshold.

The default optimisation algorithm is "BFGS", which requires a finite negative log-likelihood function evaluation finitelik=TRUE. For invalid parameters, a zero likelihood is replaced with exp(-1e6). The "BFGS" optimisation algorithms require finite values for likelihood, so any user input for finitelik will be overridden and set to finitelik=TRUE if either of these optimisation methods is chosen.

It will display a warning for non-zero convergence result comes from optim function call.

If the hessian is of reduced rank then the variance covariance (from inverse hessian) and standard error of parameters cannot be calculated, then by default std.err=TRUE and the function will stop. If you want the parameter estimates even if the hessian is of reduced rank (e.g. in a simulation study) then set std.err=FALSE.

Value

ldwm gives (log-)likelihood and nldwm gives the negative log-likelihood. fdwm returns a simple list with the following elements

call:	optim call
x:	data vector x
init:	pvector
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
wshape:	MLE of Weibull shape
wscale:	MLE of Weibull scale
mu:	MLE of Cauchy location
tau:	MLE of Cauchy scale
sigmau:	MLE of GPD scale
xi:	MLE of GPD shape

The output list has some duplicate entries and repeats some of the inputs to both provide similar items to those from fpot and to make it as useable as possible.

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd.

Note

Unlike most of the distribution functions for the extreme value mixture models, the MLE fitting only permits single scalar values for each parameter and phiu. Only the data is a vector.

When pvector=NULL then the initial values are calculated, type fdwm to see the default formulae used. The mixture model fitting can be ***extremely*** sensitive to the initial values, so you if you get a poor fit then try some alternatives. Avoid setting the starting value for the shape parameter to xi=0 as depending on the optimisation method it may be get stuck.

Infinite and missing sample values are dropped.

Error checking of the inputs is carried out and will either stop or give warning message as appropriate.

Author(s)

Yang Hu and Carl Scarrott [e-mail]

References

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

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

Frigessi, A., O. Haug, and H. Rue (2002). A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes 5 (3), 219-235

See Also

fgpd and gpd

Other fdwm: dwm

Examples

## Not run: 
set.seed(1)
par(mfrow = c(1, 1))

x = rweibull(1000, shape = 2)
xx = seq(-0.1, 4, 0.01)
y = dweibull(xx, shape = 2)

fit = fdwm(x, std.err = FALSE)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 4))
lines(xx, y)
with(fit, lines(xx, ddwm(xx, wshape, wscale, cmu, ctau, sigmau, xi), col="red"))

## End(Not run)