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
.
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)
quantiles
mean on log scale
standard deviation on log scale (positive)
threshold
scale parameter (positive)
shape parameter
probability of being above threshold TRUE
logical, if TRUE then log density
quantiles
logical, if FALSE then upper tail probabilities
cumulative probabilities
sample size (positive integer)
dlognormgpd
gives the density,
plognormgpd
gives the cumulative distribution function,
qlognormgpd
gives the quantile function and
rlognormgpd
gives a random sample.
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 phiu=TRUE
the tail fraction is estimated as the tail fraction from the
log-normal bulk model.
The cumulative distribution function with tail fraction phiu=TRUE
), upto the
threshold plnorm(x, lnmean, lnsd)
and
pgpd(x, u, sigmau, xi)
) respectively.
The cumulative distribution function for pre-specified
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.
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.
Other lognormgpd lognormgpdcon normgpd normgpdcon flognormgpd flognormgpdcon fnormgpd fnormgpdcon: lognormgpdcon
# 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 {
# }
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