mgammagpdcon: Mixture of Gammas Bulk and GPD Tail Extreme Value Mixture...
In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation

Description Usage Arguments Details Value Acknowledgments Note Author(s) References See Also Examples

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.

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)

`x`	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)
`u`	threshold
`xi`	shape parameter
`phiu`	probability of being above threshold [0, 1] or `TRUE`
`log`	logical, if TRUE then log density
`q`	quantiles
`lower.tail`	logical, if FALSE then upper tail probabilities
`p`	cumulative probabilities
`n`	sample size (positive integer)

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 φ_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 φ_u defined by the upper tail fraction of the mixture of gammas bulk model (phiu=TRUE), upto the threshold 0 < x ≤ 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 φ_u, upto the threshold 0 < x ≤ u, is given by:

F(x) = (1 - φ_u) H(x)/H(u)

and above the threshold x > u:

F(x) = φ_u + [1 - φ_u] G(x)

Notice that these definitions are equivalent when φ_u = 1 - H(u).

The continuity constraint means that (1 - φ_u) h(u)/H(u) = φ_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:

σ_u = φ_u H(u) / [1 - φ_u] h(u)

. In the special case of where the tail fraction is defined by the bulk model this reduces to

σ_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 (α):

f(0+)=∞ for 0<α<1;
f(0+)=1/β for α=1 (exponential);
f(0+)=0 for α>1;

where β is the scale parameter.

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

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

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

All inputs are vectorised except log and lower.tail, and the gamma mixture parameters can be vectorised within the list. The main inputs (x, p or q) and parameters must be either a scalar or a vector. If vectors are provided they must all be of the same length, and the function will be evaluated for each element of vector. In the case of rmgammagpdcon any input vector must be of length n.

Default values are provided for all inputs, except for the fundamentals x, q and p. The default sample size for rmgammagpdcon is 1.

Missing (NA) and Not-a-Number (NaN) values in x, p and q are passed through as is and infinite values are set to NA. None of these are not permitted for the parameters.

Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.

Carl Scarrott carl.scarrott@canterbury.ac.nz

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.

gpd and dgamma

Other gammagpdcon: fgammagpdcon, fgammagpd, fmgammagpdcon, gammagpdcon, gammagpd

Other mgamma: fmgammagpdcon, fmgammagpd, fmgamma, mgammagpd, mgamma

Other mgammagpd: fgammagpd, fmgammagpdcon, fmgammagpd, fmgamma, gammagpd, mgammagpd, mgamma

Other mgammagpdcon: fgammagpdcon, fmgammagpdcon, fmgammagpd, fmgamma, gammagpdcon, mgammagpd, mgamma

Other fmgammagpdcon: fmgammagpdcon

## 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)

## End(Not run)