lbetagpd: Log-likelihood of beta Bulk and GPD Tail Extreme Value Mixture Model

Description

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

Usage

lbetagpd(x, bshape1 = 1, bshape2 = 1,
    u = qbeta(0.9, bshape1, bshape2),
    sigmau = sqrt(bshape1 * bshape2/(bshape1 + bshape2)^2/(bshape1 + bshape2 + 1)),
    xi = 0, phiu = TRUE, log = TRUE)

  nlbetagpd(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 (bshape1, bshape2, u, sigmau, xi) or NULL

finitelik

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

bshape1

beta shape 1 (non-negative)

bshape2

beta shape 2 (non-negative)

threshold over $(0,1)$

sigmau

scale parameter (non-negative)

shape parameter

log

logical, if TRUE then log density

Value

lbetagpd gives (log-)likelihood and nlbetagpd gives the negative log-likelihood.

Details

The likelihood functions for the extreme value mixture model with beta bulk and GPD tail, as used in the maximum likelihood fitting function fbetagpd. Non-positive data are ignored. Values above 1 must come from GPD component, as threshold

u<1< code="">..

  They are designed to be used for MLE in
  fbetagpd but are available
  for wider usage, e.g. constructing your own extreme value
  mixture models.

  See fbetagpd and
  fgpd for full details.

  Log-likelihood calculations are carried out in
  lbetagpd, which takes
  parameters as inputs in the same form as distribution
  functions. The negative log-likelihood is a wrapper for
  lbetagpd, 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
  betagpd 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 beta 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 lbetagpd
  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/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