gammagpdcon: Gamma Bulk and GPD Tail Extreme Value Mixture Model with...
In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation

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

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

dgammagpdcon(x, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), xi = 0, phiu = TRUE, log = FALSE)

pgammagpdcon(q, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), xi = 0, phiu = TRUE, lower.tail = TRUE)

qgammagpdcon(p, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), xi = 0, phiu = TRUE, lower.tail = TRUE)

rgammagpdcon(n = 1, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), xi = 0, phiu = TRUE)

`x`	quantiles
`gshape`	gamma shape (positive)
`gscale`	gamma scale (positive)
`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 gamma distribution 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 gamma bulk model.

The cumulative distribution function with tail fraction φ_u defined by the upper tail fraction of the gamma 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 gamma and conditional GPD cumulative distribution functions (i.e. pgamma(x, gshape, 1/gscale) and pgpd(x, u, sigmau, xi)) respectively.

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 gamma and conditional GPD density functions (i.e. dgammma(x, gshape, gscale) and dgpd(x, u, sigmau, xi)) 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 gpd for details of GPD upper tail component and dgamma for details of gamma bulk component.

dgammagpdcon gives the density, pgammagpdcon gives the cumulative distribution function, qgammagpdcon gives the quantile function and rgammagpdcon gives a random sample.

All inputs are vectorised except log and lower.tail. 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 rgammagpdcon 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 rgammagpdcon 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.

Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz

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

Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling. 4(3), 227-244.

gpd and dgamma

Other gammagpd: fgammagpdcon, fgammagpd, fmgammagpd, fmgamma, gammagpd, mgammagpd

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

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

Other fgammagpdcon: fgammagpdcon

## Not run: 
set.seed(1)
par(mfrow = c(2, 2))

x = rgammagpdcon(1000, gshape = 2)
xx = seq(-1, 10, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dgammagpdcon(xx, gshape = 2))

# three tail behaviours
plot(xx, pgammagpdcon(xx, gshape = 2), type = "l")
lines(xx, pgammagpdcon(xx, gshape = 2, xi = 0.3), col = "red")
lines(xx, pgammagpdcon(xx, gshape = 2, xi = -0.3), col = "blue")
legend("bottomright", paste("xi =",c(0, 0.3, -0.3)),
  col=c("black", "red", "blue"), lty = 1)

x = rgammagpdcon(1000, gshape = 2, u = 3, phiu = 0.2)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dgammagpdcon(xx, gshape = 2, u = 3, phiu = 0.2))

plot(xx, dgammagpdcon(xx, gshape = 2, u = 3, xi=0, phiu = 0.2), type = "l")
lines(xx, dgammagpdcon(xx, gshape = 2, u = 3, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dgammagpdcon(xx, gshape = 2, u = 3, xi=0.2, phiu = 0.2), col = "blue")
legend("topright", c("xi = 0", "xi = 0.2", "xi = -0.2"),
  col=c("black", "red", "blue"), lty = 1)

## End(Not run)