evmix (version 1.0)

normgpdcon: Normal Bulk and GPD Tail Extreme Value Mixture Model with a 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 a continuity constraint. 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

quantile

nmean

normal mean

nsd

normal standard deviation (non-negative)

u

threshold

xi

shape parameter

phiu

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

log

logical, if TRUE then log density

q

quantile

lower.tail

logical, if FALSE then upper tail probabilities

p

cumulative probability

n

sample size (non-negative integer)

Value

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

Details

Extreme value mixture model combining normal distribution for the bulk below the threshold and GPD for upper tail with a continuity constraint. 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)). 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)). 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.

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, dnorm and dnormgpd Other normgpdcon: fnormgpdcon, lnormgpdcon, nlnormgpdcon

Examples

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)