evmix (version 1.0)

weibullgpd: Weibull 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 Weibull for bulk distribution upto the threshold and conditional GPD above threshold. The parameters are the Weibull shape wshape and scale wscale, threshold u GPD scale sigmau and shape xi and tail fraction phiu.

Usage

dweibullgpd(x, wshape = 1, wscale = 1,
    u = qweibull(0.9, wshape, wscale),
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, phiu = TRUE, log = FALSE)

  pweibullgpd(q, wshape = 1, wscale = 1,
    u = qweibull(0.9, wshape, wscale),
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, phiu = TRUE, lower.tail = TRUE)

  qweibullgpd(p, wshape = 1, wscale = 1,
    u = qweibull(0.9, wshape, wscale),
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, phiu = TRUE, lower.tail = TRUE)

  rweibullgpd(n = 1, wshape = 1, wscale = 1,
    u = qweibull(0.9, wshape, wscale),
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, phiu = TRUE)

Arguments

wshape

Weibull shape (non-negative)

wscale

Weibull scale (non-negative)

u

threshold (non-negative)

x

quantile

sigmau

scale parameter (non-negative)

xi

shape parameter

phiu

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

log

logical, if TRUE then log density

q

quantile

lower.tail

logical, if FALSE then upper tail probabilities

p

cumulative probability

n

sample size (non-negative integer)

Value

• dweibullgpd gives the density, pweibullgpd gives the cumulative distribution function, qweibullgpd gives the quantile function and rweibullgpd gives a random sample.

Details

Extreme value mixture model combining Weibull 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 Weibull bulk model. The cumulative distribution function with tail fraction $\phi_u$ defined by the upper tail fraction of the Weibull bulk model (phiu=TRUE), upto the threshold $0 \le x \le u$, 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 Weibull and conditional GPD cumulative distribution functions (i.e. pweibull(x, wshape, wscale) and pgpd(x, u, sigmau, xi)). The cumulative distribution function for pre-specified $\phi_u$, upto the threshold $0 \le x \le u$, 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 $\phi_u = 1 - H(u)$. The gamma is defined on the positive 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.

See Also

gpd and dweibull Other weibullgpd: fweibullgpd, lweibullgpd, nlweibullgpd

Examples

par(mfrow=c(2,2))
x = rweibullgpd(1000)
xx = seq(-1, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 6))
lines(xx, dweibullgpd(xx))

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

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

plot(xx, dweibullgpd(xx, xi=0, phiu = 0.2), type = "l")
lines(xx, dweibullgpd(xx, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dweibullgpd(xx, xi=0.2, phiu = 0.2), col = "blue")
legend("topleft", c("xi = 0", "xi = 0.2", "xi = -0.2"),
  col=c("black", "red", "blue"), lty = 1)