evmix (version 2.11)

gammagpdcon: 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)

Value

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

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 ${\varphi }_u$. 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 ${\varphi }_u$ 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 ${\varphi }_u$, upto the threshold $0<x\le u$, is given by: $$F(x)=(1-{\varphi }_u)H(x)/H(u)$$ and above the threshold $x>u$: $$F(x)={\varphi }_u+[1-{\varphi }_u]G(x)$$ Notice that these definitions are equivalent when ${\varphi }_u=1-H(u)$.

The continuity constraint means that $(1-{\varphi }_u)h(u)/H(u)={\varphi }_{u}g(u)$ 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: $${\sigma }_u={\varphi }_{u}H(u)/[1-{\varphi }_u]h(u)$$. 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 ($\alpha$):

• $f(0+)=\mathrm{\infty }$ 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.

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 mgamma fmgamma gammagpd gammagpdcon fgammagpd fgammagpdcon normgpd fnormgpd mgammagpd mgammagpdcon fmgammagpd fmgammagpdcon: fgammagpdcon, fgammagpd, fmgammagpdcon, fmgammagpd, fmgamma, gammagpd, mgammagpdcon, mgammagpd, mgamma

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)
# }
# NOT RUN {
# }