normgpd: 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 normal for bulk distribution upto the threshold and conditional GPD above threshold. The parameters are the normal mean nmean and standard deviation nsd, threshold u GPD scale sigmau and shape xi and tail fraction phiu.

Usage

dnormgpd(x, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  sigmau = nsd, xi = 0, phiu = TRUE, log = FALSE)
pnormgpd(q, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  sigmau = nsd, xi = 0, phiu = TRUE, lower.tail = TRUE)
qnormgpd(p, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  sigmau = nsd, xi = 0, phiu = TRUE, lower.tail = TRUE)
rnormgpd(n = 1, nmean = 0, nsd = 1, u = qnorm(0.9, nmean, nsd),
  sigmau = nsd, xi = 0, phiu = TRUE)

Arguments

quantiles

nmean

normal mean

nsd

normal standard deviation (positive)

threshold

sigmau

scale parameter (positive)

shape parameter

phiu

probability of being above threshold $[0, 1]$ or TRUE

log

logical, if TRUE then log density

quantiles

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probabilities

sample size (positive integer)

Value

dnormgpd gives the density, pnormgpd gives the cumulative distribution function, qnormgpd gives the quantile function and rnormgpd gives a random sample.

Details

Extreme value mixture model combining 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 $ϕ_{u}$ . 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 $ϕ_{u}$ defined by the upper tail fraction of the normal bulk model (phiu=TRUE), upto the threshold $x \leq u$ , 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 $ϕ_{u}$ , upto the threshold $x \leq u$ , is given by: $F (x) = (1 - ϕ_{u}) H (x) / H (u)$ and above the threshold $x > u$ : $F (x) = ϕ_{u} + [1 - ϕ_{u}] G (x)$ Notice that these definitions are equivalent when $ϕ_{u} = 1 - H (u)$ .

See gpd for details of GPD upper tail component and dnorm for details of normal bulk component.

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

Hu Y. and Scarrott, C.J. (2018). evmix: An R Package for Extreme Value Mixture Modeling, Threshold Estimation and Boundary Corrected Kernel Density Estimation. Journal of Statistical Software 84(5), 1-27. doi: 10.18637/jss.v084.i05.

Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling. 4(3), 227-244.

Examples

Run this code

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

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

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

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

plot(xx, dnormgpd(xx, xi=0, phiu = 0.2), type = "l")
lines(xx, dnormgpd(xx, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dnormgpd(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)
# }
# NOT RUN {
# }

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