R: Gamma Bulk and GPD Tail Extreme Value Mixture Model

gammagpd {evmix}

R Documentation

Gamma 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 gamma 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

dgammagpd(x, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), sigmau = sqrt(gshape) * gscale, xi = 0, phiu = TRUE,
  log = FALSE)

pgammagpd(q, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), sigmau = sqrt(gshape) * gscale, xi = 0, phiu = TRUE,
  lower.tail = TRUE)

qgammagpd(p, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), sigmau = sqrt(gshape) * gscale, xi = 0, phiu = TRUE,
  lower.tail = TRUE)

rgammagpd(n = 1, gshape = 1, gscale = 1, u = qgamma(0.9, gshape,
  1/gscale), sigmau = sqrt(gshape) * gscale, xi = 0, phiu = TRUE)

Arguments

`x`	quantiles
`gshape`	gamma shape (positive)
`gscale`	gamma scale (positive)
`u`	threshold
`sigmau`	scale parameter (positive)
`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)

Details

Extreme value mixture model combining gamma distribution for the bulk below the threshold and GPD for upper tail.

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 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 \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 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 gpd for details of GPD upper tail component and dgamma for details of gamma bulk component.

Value

dgammagpd gives the density, pgammagpd gives the cumulative distribution function, qgammagpd gives the quantile function and rgammagpd gives a random sample.

Note

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 rgammagpd 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 rgammagpd 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.

Author(s)

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

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

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

Examples

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

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

# three tail behaviours
plot(xx, pgammagpd(xx, gshape = 2), type = "l")
lines(xx, pgammagpd(xx, gshape = 2, xi = 0.3), col = "red")
lines(xx, pgammagpd(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 = rgammagpd(1000, gshape = 2, u = 3, phiu = 0.2)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dgammagpd(xx, gshape = 2, u = 3, phiu = 0.2))

plot(xx, dgammagpd(xx, gshape = 2, u = 3, xi=0, phiu = 0.2), type = "l")
lines(xx, dgammagpd(xx, gshape = 2, u = 3, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dgammagpd(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)

[Package evmix version 2.12 Index]