itmweibullgpd: Weibull Bulk and GPD Tail Interval Transition Mixture Model

Description

Density, cumulative distribution function, quantile function and random number generation for the Weibull bulk and GPD tail interval transition mixture model. The parameters are the Weibull shape wshape and scale wscale, threshold u, interval half-width epsilon, GPD scale sigmau and shape xi.

Usage

ditmweibullgpd(x, wshape = 1, wscale = 1, epsilon = sqrt(wscale^2 *
  gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
  u = qweibull(0.9, wshape, wscale), sigmau = sqrt(wscale^2 * gamma(1 +
  2/wshape) - (wscale * gamma(1 + 1/wshape))^2), xi = 0, log = FALSE)
pitmweibullgpd(q, wshape = 1, wscale = 1, epsilon = sqrt(wscale^2 *
  gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
  u = qweibull(0.9, wshape, wscale), sigmau = sqrt(wscale^2 * gamma(1 +
  2/wshape) - (wscale * gamma(1 + 1/wshape))^2), xi = 0,
  lower.tail = TRUE)
qitmweibullgpd(p, wshape = 1, wscale = 1, epsilon = sqrt(wscale^2 *
  gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
  u = qweibull(0.9, wshape, wscale), sigmau = sqrt(wscale^2 * gamma(1 +
  2/wshape) - (wscale * gamma(1 + 1/wshape))^2), xi = 0,
  lower.tail = TRUE)
ritmweibullgpd(n = 1, wshape = 1, wscale = 1,
  epsilon = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 +
  1/wshape))^2), u = qweibull(0.9, wshape, wscale),
  sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 +
  1/wshape))^2), xi = 0)

Arguments

quantiles

wshape

Weibull shape (positive)

wscale

Weibull scale (positive)

epsilon

interval half-width

threshold

sigmau

scale parameter (positive)

shape parameter

log

logical, if TRUE then log density

quantiles

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probabilities

sample size (positive integer)

Value

ditmweibullgpd gives the density, pitmweibullgpd gives the cumulative distribution function, qitmweibullgpd gives the quantile function and ritmweibullgpd gives a random sample.

Details

The interval transition mixture model combines a Weibull for the bulk model with GPD for the tail model, with a smooth transition over the interval $(u-epsilon, u+epsilon)$. The mixing function warps the Weibull to map from $(u-epsilon, u)$ to $(u-epsilon, u+epsilon)$ and warps the GPD from $(u, u+epsilon)$ to $(u-epsilon, u+epsilon)$.

The cumulative distribution function is defined by $$F(x)=\kappa(H_t(q(x)) + G(p(x)))$$ where $H_t(x)$ and $G(X)$ are the truncated Weibull and conditional GPD cumulative distribution functions (i.e. pweibull(x, wshape, wscale) and pgpd(x, u, sigmau, xi)) respectively. The truncated Weibull is not renormalised to be proper, so $H_t(x)$ contrubutes pweibull(u, wshape, wscale) to the cdf for all $x\geq (u + \epsilon)$. The normalisation constant $\kappa$ ensures a proper density, given by 1/(1+pweibull(u, wshape, wscale)) where 1 is from GPD component and latter is contribution from Weibull component.

The mixing functions $q(x)$ and $p(x)$ suggested by Holden and Haug (2013) have been implemented. These are symmetric about the threshold $u$. So for computational convenience only $q(x;u)$ has been implemented as qmix for a given $u$, with the complementary mixing function is then defined as $p(x;u)=-q(-x;-u)$.

A minor adaptation of the mixing function has been applied. For the mixture model to function correctly $q(x)>=u$ for all $x\ge u+\epsilon$, as then the bulk model will contribute the constant $H_t(u)=H(u)$ for all $x$ above the interval. Holden and Haug (2013) define $q(x)=x-\epsilon$ for all $x\ge u$. For more straightforward and interpretable computational implementation the mixing function has been set to the threshold $q(x)=u$ for all $x\ge u$, so the cdf/pdf of the Weibull model can be used directly. We do not have to define cdf/pdf for the non-proper truncated Weibull seperately. As such $q'(x)=0$ for all $x\ge u$ in qmixxprime, which also it makes clearer that Weibull does not contribute to the tail above the interval and vice-versa.

The quantile function within the transition interval is not available in closed form, so has to be solved numerically. Outside of the interval, the quantile are obtained from the Weibull and GPD components directly.

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

Holden, L. and Haug, O. (2013). A mixture model for unsupervised tail estimation. arxiv:0902.4137

Examples

Run this code

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

xx = seq(0.001, 5, 0.01)
u = 1.5
epsilon = 0.4
kappa = 1/(1 + pweibull(u, 2, 1))

f = ditmweibullgpd(xx, wshape = 2, wscale = 1, epsilon, u, sigmau = 1, xi = 0.5)
plot(xx, f, ylim = c(0, 1), xlim = c(0, 5), type = 'l', lwd = 2, xlab = "x", ylab = "density")
lines(xx, kappa * dgpd(xx, u, sigmau = 1, xi = 0.5), col = "red", lty = 2, lwd = 2)
lines(xx, kappa * dweibull(xx, 2, 1), col = "blue", lty = 2, lwd = 2)
abline(v = u + epsilon * seq(-1, 1), lty = c(2, 1, 2))
legend('topright', c('Weibull-GPD ITM', 'kappa*Weibull', 'kappa*GPD'),
      col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)

# cdf contributions
F = pitmweibullgpd(xx, wshape = 2, wscale = 1, epsilon, u, sigmau = 1, xi = 0.5)
plot(xx, F, ylim = c(0, 1), xlim = c(0, 5), type = 'l', lwd = 2, xlab = "x", ylab = "cdf")
lines(xx[xx > u], kappa * (pweibull(u, 2, 1) + pgpd(xx[xx > u], u, sigmau = 1, xi = 0.5)),
     col = "red", lty = 2, lwd = 2)
lines(xx[xx <= u], kappa * pweibull(xx[xx <= u], 2, 1), col = "blue", lty = 2, lwd = 2)
abline(v = u + epsilon * seq(-1, 1), lty = c(2, 1, 2))
legend('topright', c('Weibull-GPD ITM', 'kappa*Weibull', 'kappa*GPD'),
      col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)

# simulated data density histogram and overlay true density 
x = ritmweibullgpd(10000, wshape = 2, wscale = 1, epsilon, u, sigmau = 1, xi = 0.5)
hist(x, freq = FALSE, breaks = seq(0, 1000, 0.1), xlim = c(0, 5))
lines(xx, ditmweibullgpd(xx, wshape = 2, wscale = 1, epsilon, u, sigmau = 1, xi = 0.5),
  lwd = 2, col = 'black')  
# }
# NOT RUN {
# }

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