evmix (version 1.0)

fbetagpd: MLE Fitting of Beta Bulk and GPD Tail Extreme Value Mixture Model

Description

Maximum likelihood estimation for fitting the extreme value mixture model with beta for bulk distribution upto the threshold and conditional GPD above threshold

Usage

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

Arguments

pvector

vector of initial values mixture model parameters (bshape1, bshape2, u, sigmau, xi) or NULL

phiu

logical

control

optimisation control list (see optim)

x

vector of sample data

std.err

logical, should standard errors be calculated

method

optimisation method (see optim)

finitelik

logical, should log-likelihood return finite value for invalid parameters

...

optional inputs passed to optim

Value

• Returns a simple list with the following elements ll{ call: optim call x: data vector x init: pvector optim: complete optim output mle: vector of MLE of model parameters cov: variance-covariance matrix of MLE of model parameters se: vector of standard errors of MLE of model parameters rate: phiu to be consistent with evd nllh: minimum negative log-likelihood allparams: vector of MLE of model parameters and phiu allse: vector of standard error of all parameters and phiu n: total sample size bshape1: MLE of beta shape 1 bshape2: MLE of beta shape 2 u: threshold sigmau: MLE of GPD scale xi: MLE of GPD shape phiu: MLE of tail fraction } 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.

Details

The extreme value mixture model with beta bulk and GPD tail is fitted to the entire dataset using maximum likelihood estimation. The estimated parameters, variance-covariance matrix and their standard errors are automatically output. Non-positive data are ignored. Values above 1 must come from GPD component, as threshold u<1< code="">. The default value for phiu=TRUE so that the tail fraction is specified by beta distribution $\phi_u = 1 - H(u)$. When phiu=FALSE then the tail fraction is treated as an extra parameter estimated using the MLE which is the sample proportion above the threshold. In this case the standard error for phiu is estimated and output as sephiu. 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.

References

http://en.wikipedia.org/wiki/beta_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 MacDonald, A. (2012). Extreme value mixture modelling with medical and industrial applications. PhD thesis, University of Canterbury, New Zealand. http://ir.canterbury.ac.nz/bitstream/10092/6679/1/thesis_fulltext.pdf

See Also

lgpd and gpd Other betagpd: betagpd, dbetagpd, lbetagpd, nlbetagpd, pbetagpd, qbetagpd, rbetagpd

Examples

par(mfrow=c(2,1))
x = rbeta(1000, shape1 = 0.5, shape2 = 2)
xx = seq(-0.1, 2, 0.01)
y = dbeta(xx, shape1 = 0.5, shape2 = 2)

# Bulk model base tail fraction
fit = fbetagpd(x, phiu = TRUE, std.err = FALSE)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 2))
lines(xx, y)
lines(xx, dbetagpd(xx, bshape1 = fit$bshape1, bshape2 = fit$bshape2, u = fit$u,
  sigmau = fit$sigmau, xi = fit$xi, phiu = TRUE), col="red")
abline(v = fit$u)

# Parameterised tail fraction
fit2 = fbetagpd(x, phiu = FALSE, std.err = FALSE)
plot(xx, y, type = "l")
lines(xx, dbetagpd(xx, bshape1 = fit$bshape1, bshape2 = fit$bshape2,, u = fit$u,
  sigmau = fit$sigmau, xi = fit$xi, phiu = TRUE), col="red")
lines(xx, dbetagpd(xx, bshape1 = fit2$bshape1, bshape2 = fit2$bshape2,, u = fit2$u,
  sigmau = fit2$sigmau, xi = fit2$xi, phiu = fit2$phiu), col="blue")
abline(v = fit$u, col = "red")
abline(v = fit2$u, col = "blue")
legend("topright", c("True Density","Bulk Tail Fraction","Parameterised Tail Fraction"),
  col=c("black", "red", "blue"), lty = 1)