evmix (version 2.11)

betagpd: Beta 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 beta for bulk distribution upto the threshold and conditional GPD above threshold. The parameters are the beta shape 1 bshape1 and shape 2 bshape2, threshold u GPD scale sigmau and shape xi and tail fraction phiu.

Usage

dbetagpd(x, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1, bshape2),
  sigmau = sqrt(bshape1 * bshape2/(bshape1 + bshape2)^2/(bshape1 + bshape2 +
  1)), xi = 0, phiu = TRUE, log = FALSE)
pbetagpd(q, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1, bshape2),
  sigmau = sqrt(bshape1 * bshape2/(bshape1 + bshape2)^2/(bshape1 + bshape2 +
  1)), xi = 0, phiu = TRUE, lower.tail = TRUE)
qbetagpd(p, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1, bshape2),
  sigmau = sqrt(bshape1 * bshape2/(bshape1 + bshape2)^2/(bshape1 + bshape2 +
  1)), xi = 0, phiu = TRUE, lower.tail = TRUE)
rbetagpd(n = 1, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1,
  bshape2), sigmau = sqrt(bshape1 * bshape2/(bshape1 + bshape2)^2/(bshape1 +
  bshape2 + 1)), xi = 0, phiu = TRUE)

Arguments

x

quantiles

bshape1

beta shape 1 (positive)

bshape2

beta shape 2 (positive)

u

threshold over $(0,1)$

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)

Value

dbetagpd gives the density, pbetagpd gives the cumulative distribution function, qbetagpd gives the quantile function and rbetagpd gives a random sample.

Details

Extreme value mixture model combining beta 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 ${\varphi }_u$. Alternatively, when phiu=TRUE the tail fraction is estimated as the tail fraction from the beta bulk model.

The usual beta distribution is defined over $[0,1]$, but this mixture is generally not limited in the upper tail $[0,\mathrm{\infty }]$, except for the usual upper tail limits for the GPD when xi<0 discussed in gpd. Therefore, the threshold is limited to $(0,1)$.

The cumulative distribution function with tail fraction ${\varphi }_u$ defined by the upper tail fraction of the beta bulk model (phiu=TRUE), upto the threshold $0\le x\le u<1$, 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 beta and conditional GPD cumulative distribution functions (i.e. pbeta(x, bshape1, bshape2) and pgpd(x, u, sigmau, xi)).

The cumulative distribution function for pre-specified ${\varphi }_u$, upto the threshold $0\le x\le u<1$, 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 ${\varphi }_u=1-H(u)$.

See gpd for details of GPD upper tail component and dbeta for details of beta bulk component.

References

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

MacDonald, A. (2012). Extreme value mixture modelling with medical and industrial applications. PhD thesis, University of Canterbury, New Zealand. http://ir.canterbury.ac.nz/bitstream/10092/6679/1/thesis_fulltext.pdf

See Also

gpd and dbeta

Other betagpd betagpdcon fbetagpd fbetagpdcon: betagpdcon

Examples

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

x = rbetagpd(1000, bshape1 = 1.5, bshape2 = 2, u = 0.7, phiu = 0.2)
xx = seq(-0.1, 2, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 2))
lines(xx, dbetagpd(xx, bshape1 = 1.5, bshape2 = 2, u = 0.7, phiu = 0.2))

# three tail behaviours
plot(xx, pbetagpd(xx, bshape1 = 1.5, bshape2 = 2, u = 0.7, phiu = 0.2), type = "l")
lines(xx, pbetagpd(xx, bshape1 = 1.5, bshape2 = 2, u = 0.7, phiu = 0.2, xi = 0.3), col = "red")
lines(xx, pbetagpd(xx, bshape1 = 1.5, bshape2 = 2, u = 0.7, phiu = 0.2, xi = -0.3), col = "blue")
legend("bottomright", paste("xi =",c(0, 0.3, -0.3)),
  col=c("black", "red", "blue"), lty = 1)

x = rbetagpd(1000, bshape1 = 2, bshape2 = 0.8, u = 0.7, phiu = 0.5)
hist(x, breaks = 100, freq = FALSE, xlim = c(-0.1, 2))
lines(xx, dbetagpd(xx, bshape1 = 2, bshape2 = 0.6, u = 0.7, phiu = 0.5))

plot(xx, dbetagpd(xx, bshape1 = 2, bshape2 = 0.8, u = 0.7, phiu = 0.5, xi=0), type = "l")
lines(xx, dbetagpd(xx, bshape1 = 2, bshape2 = 0.8, u = 0.7, phiu = 0.5, xi=-0.2), col = "red")
lines(xx, dbetagpd(xx, bshape1 = 2, bshape2 = 0.8, u = 0.7, phiu = 0.5, xi=0.2), col = "blue")
legend("topright", c("xi = 0", "xi = 0.2", "xi = -0.2"),
  col=c("black", "red", "blue"), lty = 1)
# }
# NOT RUN {
# }