mgammagpdcon: Mixture of Gammas Bulk and GPD Tail Extreme Value Mixture Model with Single Continuity Constraint

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 for upper tail with continuity at threshold. The parameters are the multiple gamma shapes mgshape, scales mgscale and mgweights, threshold u GPD shape xi and tail fraction phiu.

Usage

dmgammagpdcon(x, mgshape = 1, mgscale = 1, mgweight = NULL,
  u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]), xi = 0, phiu = TRUE,
  log = FALSE)
pmgammagpdcon(q, mgshape = 1, mgscale = 1, mgweight = NULL,
  u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]), xi = 0, phiu = TRUE,
  lower.tail = TRUE)
qmgammagpdcon(p, mgshape = 1, mgscale = 1, mgweight = NULL,
  u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]), xi = 0, phiu = TRUE,
  lower.tail = TRUE)
rmgammagpdcon(n = 1, mgshape = 1, mgscale = 1, mgweight = NULL,
  u = qgamma(0.9, mgshape[[1]], 1/mgscale[[1]]), xi = 0, phiu = TRUE)

Arguments

quantiles

mgshape

mgamma shape (positive) as list or vector

mgscale

mgamma scale (positive) as list or vector

mgweight

mgamma weights (positive) as list or vector (NULL for equi-weighted)

threshold

shape parameter

phiu

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

log

logical, if TRUE then log density

quantiles

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probabilities

sample size (positive integer)

Value

dmgammagpdcon gives the density, pmgammagpdcon gives the cumulative distribution function, qmgammagpdcon gives the quantile function and rmgammagpdcon gives a random sample.

Acknowledgments

Thanks to Daniela Laas, University of St Gallen, Switzerland for reporting various bugs in these functions.

Details

Extreme value mixture model combining mixture of gammas for the bulk below the threshold and GPD for upper tail with continuity at threshold.

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 mixture of gammas bulk model.

Suppose there are $M>=1$ gamma components in the mixture model. If you wish to have a single (scalar) value for each parameter within each of the $M$ components then these can be input as a vector of length $M$. If you wish to input a vector of values for each parameter within each of the $M$ components, then they are input as a list with each entry the parameter object for each component (which can either be a scalar or vector as usual). No matter whether they are input as a vector or list there must be $M$ elements in mgshape and mgscale, one for each gamma mixture component. Further, any vectors in the list of parameters must of the same length of the x, q, p or equal to the sample size n, where relevant.

If mgweight=NULL then equal weights for each component are assumed. Otherwise, mgweight 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 cumulative distribution function with tail fraction $\phi_u$ defined by the upper tail fraction of the mixture of gammas 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.

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 continuity constraint means that $(1 - \phi_u) h(u)/H(u) = \phi_u g(u)$ where $h(x)$ and $g(x)$ are the mixture of gammas and conditional GPD density functions respectively. The resulting GPD scale parameter is then: $$\sigma_u = \phi_u H(u) / [1 - \phi_u] h(u)$$. In the special case of where the tail fraction is defined by the bulk model this reduces to $$\sigma_u = [1 - H(u)] / h(u)$$.

The gamma is defined on the non-negative reals, so the threshold must be positive. Though behaviour at zero depends on the shape ($\alpha$):

$f(0+)=\infty$ for $0<\alpha<1$;
$f(0+)=1/\beta$ for $\alpha=1$ (exponential);
$f(0+)=0$ for $\alpha>1$;

where $\beta$ is the scale parameter.

See gammagpd for details of simpler parametric mixture model with single gamma for bulk component and GPD for upper tail.

References

http://www.math.canterbury.ac.nz/~c.scarrott/evmix

http://en.wikipedia.org/wiki/Gamma_distribution

http://en.wikipedia.org/wiki/Generalized_Pareto_distribution

http://en.wikipedia.org/wiki/Mixture_model

McLachlan, G.J. and Peel, D. (2000). Finite Mixture Models. Wiley.

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.

Examples

Run this code

# NOT RUN {
set.seed(1)
par(mfrow = c(1, 1))

x = rmgammagpdcon(1000, mgshape = c(1, 6), mgscale = c(1, 2), mgweight = c(1, 2), u = 15, xi = 0)
xx = seq(-1, 40, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 40))
lines(xx, dmgammagpdcon(xx, mgshape = c(1, 6), mgscale = c(1, 2), mgweight = c(1, 2),
 u = 15, xi = 0))
abline(v = 15)
# }
# NOT RUN {
# }

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