betagpdcon function

Beta Bulk and GPD Tail Extreme Value Mixture Model with Single Continuity Constraint

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 with continuity at threshold. The parameters are the beta shape 1 bshape1 and shape 2 bshape2, threshold u

GPD shape xi and tail fraction phiu.

dbetagpdcon(x, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1,
  bshape2), xi = 0, phiu = TRUE, log = FALSE)

pbetagpdcon(q, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1,
  bshape2), xi = 0, phiu = TRUE, lower.tail = TRUE)

qbetagpdcon(p, bshape1 = 1, bshape2 = 1, u = qbeta(0.9, bshape1,
  bshape2), xi = 0, phiu = TRUE, lower.tail = TRUE)

rbetagpdcon(n = 1, bshape1 = 1, bshape2 = 1, u = qbeta(0.9,
  bshape1, bshape2), xi = 0, phiu = TRUE)

Arguments

• x: quantiles
• bshape1: beta shape 1 (positive)
• bshape2: beta shape 2 (positive)
• u: threshold over $(0, 1)$
• 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

dbetagpdcon gives the density, pbetagpdcon gives the cumulative distribution function, qbetagpdcon gives the quantile function and rbetagpdcon gives a random sample.

Details

Extreme value mixture model combining beta 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 $\phi_u$. 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 $[0,\infty]$, 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 $\phi_u$ 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 $\phi_u$, upto the threshold $0 \le x \le u < 1$, 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 continuity constraint means that $(1 - \phi_u) h(u)/H(u) = \phi_u g(u)$

where $h(x)$ and $g(x)$ are the beta and conditional GPD density functions (i.e. dbeta(x, bshape1, bshape2) and dgpd(x, u, sigmau, xi)) respectively. The resulting GPD scale parameter is then:

$$\sigma_u = \phi_u H(u) / [1 - \phi_u] h(u)$$

. In the special case of where the tail fraction is defined by the bulk model this reduces to

$$\sigma_u = [1 - H(u)] / h(u)$$

.

See gpd for details of GPD upper tail component and dbeta for details of beta 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 rbetagpdcon 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 rbetagpdcon 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 = rbetagpdcon(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, dbetagpdcon(xx, bshape1 = 1.5, bshape2 = 2, u = 0.7, phiu = 0.2))

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

x = rbetagpdcon(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, dbetagpdcon(xx, bshape1 = 2, bshape2 = 0.6, u = 0.7, phiu = 0.5))

plot(xx, dbetagpdcon(xx, bshape1 = 2, bshape2 = 0.8, u = 0.7, phiu = 0.5, xi=0), type = "l")
lines(xx, dbetagpdcon(xx, bshape1 = 2, bshape2 = 0.8, u = 0.7, phiu = 0.5, xi=-0.2), col = "red")
lines(xx, dbetagpdcon(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)
## End(Not run)

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: betagpd, fbetagpdcon, fbetagpd

Other betagpdcon: betagpd, fbetagpdcon, fbetagpd

Other fbetagpdcon: fbetagpdcon

Author(s)

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