normgpdcon {evmix}

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

Description

Density, cumulative distribution function, quantile function and random number generation for the extreme value mixture model with normal for bulk distribution upto the threshold and conditional GPD above threshold with continuity at threshold. The parameters are the normal mean nmean and standard deviation nsd, threshold u and GPD shape xi and tail fraction phiu.

Usage

dnormgpdcon(x, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  xi = 0, phiu = TRUE, log = FALSE)

pnormgpdcon(q, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  xi = 0, phiu = TRUE, lower.tail = TRUE)

qnormgpdcon(p, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  xi = 0, phiu = TRUE, lower.tail = TRUE)

rnormgpdcon(n = 1, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  xi = 0, phiu = TRUE)

Arguments

x

quantiles

nmean

normal mean

nsd

normal standard deviation (positive)

u

threshold

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)

Details

Extreme value mixture model combining normal 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 normal bulk model.

The cumulative distribution function with tail fraction \phi_u defined by the upper tail fraction of the normal bulk model (phiu=TRUE), upto the threshold 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 normal and conditional GPD cumulative distribution functions (i.e. pnorm(x, nmean, nsd) and pgpd(x, u, sigmau, xi)) respectively.

The cumulative distribution function for pre-specified \phi_u, upto the threshold 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 continuity constraint means that (1 - \phi_u) h(u)/H(u) = \phi_u g(u) where h(x) and g(x) are the normal and conditional GPD density functions (i.e. dnorm(x, nmean, nsd) 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 dnorm for details of normal bulk component.

Value

dnormgpdcon gives the density, pnormgpdcon gives the cumulative distribution function, qnormgpdcon gives the quantile function and rnormgpdcon gives a random sample.

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 rnormgpdcon 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 rnormgpdcon 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.

Due to symmetry, the lower tail can be described by GPD by negating the quantiles. The normal mean nmean and GPD threshold u will also require negation.

Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.

Author(s)

Yang Hu and Carl Scarrott [e-mail]

References

http://en.wikipedia.org/wiki/Normal_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 dnorm

Other normgpd: fgng, fhpd, fitmnormgpd, flognormgpd, fnormgpdcon, fnormgpd, gngcon, gng, hpdcon, hpd, itmnormgpd, lognormgpdcon, lognormgpd, normgpd

Other normgpdcon: fgngcon, fhpdcon, flognormgpdcon, fnormgpdcon, fnormgpd, gngcon, gng, hpdcon, hpd, normgpd

Other gngcon: fgngcon, fgng, fnormgpdcon, gngcon, gng

Other fnormgpdcon: fnormgpdcon

Examples

## Not run: 
set.seed(1)
par(mfrow = c(2, 2))

x = rnormgpdcon(1000)
xx = seq(-4, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 6))
lines(xx, dnormgpdcon(xx))

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

x = rnormgpdcon(1000, phiu = 0.2)
xx = seq(-4, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 6))
lines(xx, dnormgpdcon(xx, phiu = 0.2))

plot(xx, dnormgpdcon(xx, xi=0, phiu = 0.2), type = "l")
lines(xx, dnormgpdcon(xx, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dnormgpdcon(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)