lweibullgpd: Log-likelihood of Weibull Bulk and GPD Tail Extreme Value Mixture Model

Description

Log-likelihood and negative log-likelihood for the extreme value mixture model with Weibull for bulk distribution upto the threshold and conditional GPD above threshold.

Usage

lweibullgpd(x, wshape = 1, wscale = 1,
    u = qweibull(0.9, wshape, wscale),
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, phiu = TRUE, log = TRUE)

  nlweibullgpd(pvector, x, phiu = TRUE, finitelik = FALSE)

Arguments

phiu

probability of being above threshold [0,1] or logical

vector of sample data

pvector

vector of initial values mixture model parameters (wshape, wscale, u, sigmau, xi) or NULL

finitelik

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

wshape

Weibull shape (non-negative)

wscale

Weibull scale (non-negative)

threshold (non-negative)

sigmau

scale parameter (non-negative)

shape parameter

log

logical, if TRUE then log density

Value

lweibullgpd gives (log-)likelihood and nlweibullgpd gives the negative log-likelihood.

Details

The likelihood functions for the extreme value mixture model with Weibull bulk and GPD tail, as used in the maximum likelihood fitting function fweibullgpd. Non-positive data are ignored. They are designed to be used for MLE in fweibullgpd but are available for wider usage, e.g. constructing your own extreme value mixture models. See fweibullgpd and fgpd for full details. Log-likelihood calculations are carried out in lweibullgpd, which takes parameters as inputs in the same form as distribution functions. The negative log-likelihood is a wrapper for lweibullgpd, designed towards making it useable for optimisation (e.g. parameters are given a vector as first input). The tail fraction phiu is treated separately to the other parameters, to allow for all it's representations. Unlike the distribution functions weibullgpd the phiu must be either logical (TRUE or FALSE) or numerical in range $(0, 1)$. The default is to specify phiu=TRUE so that the tail fraction is specified by Weibull distribution $\phi_u = 1 - H(u)$, or phiu=FALSE to treat the tail fraction as an extra parameter estimated using the sample proportion. Specify a numeric phiu as pre-specified probability $(0, 1)$. Notice that the tail fraction probability cannot be 0 or 1 otherwise there would be no contribution from either tail or bulk components respectively. The function lweibullgpd carries out the calculations for the log-likelihood directly, which can be exponentiated to give actual likelihood using (log=FALSE).

References

http://en.wikipedia.org/wiki/Weibull_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 Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling. 4(3), 227-244.