lgngcon: Log-likelihood of Normal Bulk with GPD Upper and Lower Tails Extreme Value Mixture Model with Continuity Constraints

Description

Log-likelihood and negative log-likelihood for the extreme value mixture model with normal for bulk distribution between the lower and upper thresholds with conditional GPD for the two tails with continuity constraints.

Usage

lgngcon(x, nmean = 0, nsd = 1,
    ul = qnorm(0.1, nmean, nsd), xil = 0, phiul = TRUE,
    ur = qnorm(0.9, nmean, nsd), xir = 0, phiur = TRUE,
    log = TRUE)

  nlgngcon(pvector, x, phiul = TRUE, phiur = TRUE,
    finitelik = FALSE)

Arguments

vector of sample data

nmean

normal mean

nsd

normal standard deviation (non-negative)

lower tail threshold

xil

lower tail GPD shape parameter

phiul

probability of being above threshold (0, 1) or logical

upper tail threshold

xir

upper tail GPD shape parameter

phiur

probability of being above threshold (0, 1) or logical

log

logical, if TRUE then log density

pvector

vector of initial values of mixture model parameters or NULL

finitelik

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

Value

lgngcon gives (log-)likelihood and nlgngcon gives the negative log-likelihood.

Details

The likelihood functions for the extreme value mixture model with normal bulk and GPD for the two tails, as used in the maximum likelihood fitting function fgngcon. They are designed to be used for MLE in fgngcon but are available for wider usage, e.g. constructing your own extreme value mixture models. See fgngcon, gngcon and fgpd for full details. Log-likelihood calculations are carried out in lgngcon, which takes parameters as inputs in the same form as distribution functions. The negative log-likelihood is a wrapper for lgngcon, designed towards making it useable for optimisation (e.g. parameters are given a vector as first input). The tail fractions phiul and phiur are treated separately to the other parameters, to allow for all it's representations. Unlike the distribution functions gngcon 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 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 lgngcon 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/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 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.