gammagpdcon {evmix}

Gamma 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 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.

Usage

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)

Arguments

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)

Details

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 \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 continuity constraint means that (1 - \phi_u) h(u)/H(u) = \phi_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:

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

Value

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

Author(s)

Yang Hu and Carl Scarrott [e-mail]

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.

See Also

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

Examples

## 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)