weibullgpd function

Weibull Bulk and GPD Tail Extreme Value Mixture Model

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.

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

• x: quantiles
• wshape: Weibull shape (positive)
• wscale: Weibull scale (positive)
• u: threshold
• sigmau: scale parameter (positive)
• 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)

Returns

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 $0 < x \le u$, 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)) 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 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.

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 rweibullgpd 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 rweibullgpd 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.

Examples

## Not run:

set.seed(1)
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)
## End(Not run)

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: fitmweibullgpd, fweibullgpdcon, fweibullgpd, itmweibullgpd, weibullgpdcon

Other weibullgpdcon: fweibullgpdcon, fweibullgpd, itmweibullgpd, weibullgpdcon

Other itmweibullgpd: fitmweibullgpd, fweibullgpdcon, fweibullgpd, itmweibullgpd, weibullgpdcon

Other fweibullgpd: fweibullgpd

Author(s)

Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz