llognormgpd: Log-likelihood of Log-Normal Bulk and GPD Tail Extreme Value Mixture Model

Description

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

Usage

llognormgpd(x, lnmean = 0, lnsd = 1,
    u = qlnorm(0.9, lnmean, lnsd), sigmau = lnsd, xi = 0,
    phiu = TRUE, log = TRUE)

  nllognormgpd(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 of mixture model parameters (lnmean, lnsd, u, sigmau, xi) or NULL

finitelik

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

lnmean

mean on log scale

lnsd

standard deviation on log scale (non-negative)

threshold

sigmau

scale parameter (non-negative)

shape parameter

log

logical, if TRUE then log density

Value

llognormgpd gives (log-)likelihood and nllognormgpd gives the negative log-likelihood.

Details

The likelihood functions for the extreme value mixture model with log-normal bulk and GPD tail, as used in the maximum likelihood fitting function flognormgpd. They are designed to be used for MLE in flognormgpd but are available for wider usage, e.g. constructing your own extreme value mixture models. Negative data are ignored. See flognormgpd, fnormgpd and fgpd for full details. Log-likelihood calculations are carried out in llognormgpd, which takes parameters as inputs in the same form as distribution functions. The negative log-likelihood is a wrapper for llognormgpd, 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 lognormgpd 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 log-normal 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 llognormgpd 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/Log-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 Solari, S. and Losada, M.A. (2004). A unified statistical model for hydrological variables including the selection of threshold for the peak over threshold method. Water Resources Research. 48, W10541.