evmix (version 2.11)

lognormgpd: Log-Normal 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 log-normal for bulk distribution upto the threshold and conditional GPD above threshold. The parameters are the log-normal mean lnmean and standard deviation lnsd, threshold u GPD scale sigmau and shape xi and tail fraction phiu.

Usage

dlognormgpd(x, lnmean = 0, lnsd = 1, u = qlnorm(0.9, lnmean, lnsd),
  sigmau = lnsd, xi = 0, phiu = TRUE, log = FALSE)
plognormgpd(q, lnmean = 0, lnsd = 1, u = qlnorm(0.9, lnmean, lnsd),
  sigmau = lnsd, xi = 0, phiu = TRUE, lower.tail = TRUE)
qlognormgpd(p, lnmean = 0, lnsd = 1, u = qlnorm(0.9, lnmean, lnsd),
  sigmau = lnsd, xi = 0, phiu = TRUE, lower.tail = TRUE)
rlognormgpd(n = 1, lnmean = 0, lnsd = 1, u = qlnorm(0.9, lnmean, lnsd),
  sigmau = lnsd, xi = 0, phiu = TRUE)

Arguments

x

quantiles

lnmean

mean on log scale

lnsd

standard deviation on log 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)

Value

dlognormgpd gives the density, plognormgpd gives the cumulative distribution function, qlognormgpd gives the quantile function and rlognormgpd gives a random sample.

Details

Extreme value mixture model combining log-normal 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 log-normal bulk model.

The cumulative distribution function with tail fraction \(\phi_u\) defined by the upper tail fraction of the log-normal 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 log-normal and conditional GPD cumulative distribution functions (i.e. plnorm(x, lnmean, lnsd) 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 log-normal is defined on the positive reals, so the threshold must be positive.

See gpd for details of GPD upper tail component and dlnorm for details of log-normal bulk component.

References

http://en.wikipedia.org/wiki/Log-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

Solari, S. and Losada, M.A. (2004). A unified statistical model for hydrological variables including the selection of threshold for the peak over threshold method. Water Resources Research. 48, W10541.

See Also

gpd and dlnorm

Other lognormgpd lognormgpdcon normgpd normgpdcon flognormgpd flognormgpdcon fnormgpd fnormgpdcon: lognormgpdcon

Examples

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

x = rlognormgpd(1000)
xx = seq(-1, 10, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dlognormgpd(xx))

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

x = rlognormgpd(1000, u = 2, phiu = 0.2)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dlognormgpd(xx, u = 2, phiu = 0.2))

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