weibullgpdcon: Weibull 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 Weibull for bulk distribution upto the threshold and conditional GPD above threshold with continuity at threshold. The parameters are the weibull shape wshape and scale wscale, threshold u GPD shape xi and tail fraction phiu.

Usage

dweibullgpdcon(x, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
  wscale), xi = 0, phiu = TRUE, log = FALSE)
pweibullgpdcon(q, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
  wscale), xi = 0, phiu = TRUE, lower.tail = TRUE)
qweibullgpdcon(p, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
  wscale), xi = 0, phiu = TRUE, lower.tail = TRUE)
rweibullgpdcon(n = 1, wshape = 1, wscale = 1, u = qweibull(0.9,
  wshape, wscale), xi = 0, phiu = TRUE)

Arguments

quantiles

wshape

Weibull shape (positive)

wscale

Weibull scale (positive)

threshold

shape parameter

phiu

probability of being above threshold $[0, 1]$ or TRUE

log

logical, if TRUE then log density

quantiles

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probabilities

sample size (positive integer)

Value

dweibullgpdcon gives the density, pweibullgpdcon gives the cumulative distribution function, qweibullgpdcon gives the quantile function and rweibullgpdcon gives a random sample.

Acknowledgments

Thanks to Ben Youngman, Exeter University, UK for reporting a bug in the rweibullgpdcon function.

Details

Extreme value mixture model combining Weibull 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 weibull bulk model.

The cumulative distribution function with tail fraction $ϕ_{u}$ defined by the upper tail fraction of the Weibull bulk model (phiu=TRUE), upto the threshold $0 < x \leq u$ , 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 Weibull and conditional GPD cumulative distribution functions (i.e. pweibull(x, wshape, wscale) and pgpd(x, u, sigmau, xi)) respectively.

The cumulative distribution function for pre-specified $ϕ_{u}$ , upto the threshold $0 < x \leq u$ , 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 Weibull and conditional GPD density functions (i.e. dweibull(x, wshape, wscale) 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 Weibull is defined on the non-negative reals, so the threshold must be positive.

See gpd for details of GPD upper tail component and dweibull for details of weibull bulk component.

References

http://en.wikipedia.org/wiki/Weibull_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

Run this code

# NOT RUN {
set.seed(1)
par(mfrow = c(2, 2))

x = rweibullgpdcon(1000)
xx = seq(-0.1, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 6))
lines(xx, dweibullgpdcon(xx))

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

x = rweibullgpdcon(1000, phiu = 0.2)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 6))
lines(xx, dweibullgpdcon(xx, phiu = 0.2))

plot(xx, dweibullgpdcon(xx, xi=0, phiu = 0.2), type = "l")
lines(xx, dweibullgpdcon(xx, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dweibullgpdcon(xx, 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 {
# }

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