mgammagpd: Mixture of Gammas Bulk and GPD Tail Extreme Value Mixture Model

Description

Density, cumulative distribution function, quantile function and random number generation for the extreme value mixture model with mixture of gammas for bulk distribution upto the threshold and conditional GPD above threshold. The parameters are the gamma shape gshape and scale gscale, threshold u GPD scale sigmau and shape xi and tail fraction phiu.

Usage

dmgammagpd(x, mgshape = list(1), mgscale = list(1),
    mgweights = NULL,
    u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]),
    sigmau = sqrt(mgshape[[1]]) * mgscale[[1]], xi = 0,
    phiu = TRUE, log = FALSE)

  pmgammagpd(q, mgshape = list(1), mgscale = list(1),
    mgweights = NULL,
    u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]),
    sigmau = sqrt(mgshape[[1]]) * mgscale[[1]], xi = 0,
    phiu = TRUE, lower.tail = TRUE)

  qmgammagpd(p, mgshape = list(1), mgscale = list(1),
    mgweights = NULL,
    u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]),
    sigmau = sqrt(mgshape[[1]]) * mgscale[[1]], xi = 0,
    phiu = TRUE, lower.tail = TRUE)

  rmgammagpd(n = 1, mgshape = list(1), mgscale = list(1),
    mgweights = NULL,
    u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]),
    sigmau = sqrt(mgshape[[1]]) * mgscale[[1]], xi = 0,
    phiu = TRUE)

Arguments

mgshape

mgamma shape (non-negative) as list

mgscale

mgamma scale (non-negative) as list

mgweights

mgamma weights (positive) as list or NULL

quantile

threshold (non-negative)

sigmau

scale parameter (non-negative)

shape parameter

phiu

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

log

logical, if TRUE then log density

quantile

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probability

sample size (non-negative integer)

Value

dmgammagpd gives the density, pmgammagpd gives the cumulative distribution function, qmgammagpd gives the quantile function and rmgammagpd gives a random sample.

Details

Extreme value mixture model combining mixture of gammas for the bulk below the threshold and GPD for upper tail. The parameters are input as a list, with one parameter object in the list for each gamma component. There must be the same number of components in mgshape and mgscale. The number of objects in the parameters lists determines the number of components. The parameter object for each gamma component can either be a scalar or vector, consistent with the other mixture models If mgweights=NULL then assumes equal weights for each component. Otherwise, mgweights must be a list of the same length as mgshape and mgscale, filled with positive values. In the latter case, the weights are rescaled to sum to unity. The user can pre-specify phiu permitting a parameterised value for the tail fraction $\phi_u$. Alternatively, when phiu=TRUE the tail fraction is estimated as the tail fraction from the gamma bulk model. The cumulative distribution function with tail fraction $\phi_u$ defined by the upper tail fraction of the gamma bulk model (phiu=TRUE), upto the threshold $0 < x \le u$, given by: $$F(x) = H(x)$$ and above the threshold $x > u$: $$F(x) = H(u) + [1 - H(u)] G(x)$$ where $H(x)$ and $G(X)$ are the mixture of gammas and conditional GPD cumulative distribution functions respectively. The cumulative distribution function for pre-specified $\phi_u$, upto the threshold $0 < x \le u$, is given by: $$F(x) = (1 - \phi_u) H(x)/H(u)$$ and above the threshold $x > u$: $$F(x) = \phi_u + [1 - \phi_u] G(x)$$ Notice that these definitions are equivalent when $\phi_u = 1 - H(u)$. The gamma is defined on the non-negative reals, so the threshold must be non-negative. See gammagpd for details of simpler parametric mixture model with single gamma for bulk component and GPD for upper tail.

References

http://en.wikipedia.org/wiki/Gamma_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 do Nascimento, F.F., Gamerman, D. and Lopes, H.F. (2011). A semiparametric Bayesian approach to extreme value estimation. Statistical Computing, 22(2), 661-675.