itmnormgpd function

Normal Bulk and GPD Tail Interval Transition Mixture Model

Density, cumulative distribution function, quantile function and random number generation for the normal bulk and GPD tail interval transition mixture model. The parameters are the normal mean nmean and standard deviation nsd, threshold u, interval half-width epsilon, GPD scale sigmau and shape xi.

ditmnormgpd(x, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
  nmean, nsd), sigmau = nsd, xi = 0, log = FALSE)

pitmnormgpd(q, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
  nmean, nsd), sigmau = nsd, xi = 0, lower.tail = TRUE)

qitmnormgpd(p, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
  nmean, nsd), sigmau = nsd, xi = 0, lower.tail = TRUE)

ritmnormgpd(n = 1, nmean = 0, nsd = 1, epsilon = nsd,
  u = qnorm(0.9, nmean, nsd), sigmau = nsd, xi = 0)

Arguments

• x: quantiles
• nmean: normal mean
• nsd: normal standard deviation (positive)
• epsilon: interval half-width
• u: threshold
• sigmau: scale parameter (positive)
• xi: shape parameter
• 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)

Returns

ditmnormgpd gives the density, pitmnormgpd gives the cumulative distribution function, qitmnormgpd gives the quantile function and ritmnormgpd gives a random sample.

Details

The interval transition mixture model combines a normal for the bulk model with GPD for the tail model, with a smooth transition over the interval $(u-epsilon, u+epsilon)$. The mixing function warps the normal to map from $(u-epsilon, u)$ to $(u-epsilon, u+epsilon)$ and warps the GPD from $(u, u+epsilon)$ to $(u-epsilon, u+epsilon)$.

The cumulative distribution function is defined by

$$F(x)=\kappa(H_t(q(x)) + G(p(x)))$$

where $H_t(x)$ and $G(x)$ are the truncated normal and conditional GPD cumulative distribution functions (i.e. pnorm(x, nmean, nsd) and pgpd(x, u, sigmau, xi)) respectively. The truncated normal is not renormalised to be proper, so $H_t(x)$ contrubutes pnorm(u, nmean, nsd) to the cdf for all $x\geq (u + \epsilon)$. The normalisation constant $\kappa$ ensures a proper density, given by 1/(1+pnorm(u, nmean, nsd)) where 1 is from GPD component and latter is contribution from normal component.

The mixing functions $q(x)$ and $p(x)$ suggested by Holden and Haug (2013) have been implemented. These are symmetric about the threshold $u$. So for computational convenience only $q(x;u)$ has been implemented as qmix

for a given $u$, with the complementary mixing function is then defined as $p(x;u)=-q(-x;-u)$.

A minor adaptation of the mixing function has been applied. For the mixture model to function correctly $q(x)>=u$ for all $x\ge u+\epsilon$, as then the bulk model will contribute the constant $H_t(u)=H(u)$ for all $x$ above the interval. Holden and Haug (2013) define $q(x)=x-\epsilon$ for all $x\ge u$. For more straightforward and interpretable computational implementation the mixing function has been set to the threshold $q(x)=u$ for all $x\ge u$, so the cdf/pdf of the normal model can be used directly. We do not have to define cdf/pdf for the non-proper truncated normal seperately. As such $q'(x)=0$ for all $x\ge u$ in qmixxprime, which also makes it clearer that normal does not contribute to the tail above the interval and vice-versa.

The quantile function within the transition interval is not available in closed form, so has to be solved numerically. Outside of the interval, the quantile are obtained from the normal and GPD components directly.

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 ritmnormgpd 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 ritmnormgpd 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.

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

Examples

## Not run:

set.seed(1)
par(mfrow = c(2, 2))

xx = seq(-4, 5, 0.01)
u = 1.5
epsilon = 0.4
kappa = 1/(1 + pnorm(u, 0, 1))

f = ditmnormgpd(xx, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5)
plot(xx, f, ylim = c(0, 1), xlim = c(-4, 5), type = 'l', lwd = 2, xlab = "x", ylab = "density")
lines(xx, kappa * dgpd(xx, u, sigmau = 1, xi = 0.5), col = "red", lty = 2, lwd = 2)
lines(xx, kappa * dnorm(xx, 0, 1), col = "blue", lty = 2, lwd = 2)
abline(v = u + epsilon * seq(-1, 1), lty = c(2, 1, 2))
legend('topright', c('Normal-GPD ITM', 'kappa*Normal', 'kappa*GPD'),
      col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)

# cdf contributions
F = pitmnormgpd(xx, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5)
plot(xx, F, ylim = c(0, 1), xlim = c(-4, 5), type = 'l', lwd = 2, xlab = "x", ylab = "cdf")
lines(xx[xx > u], kappa * (pnorm(u, 0, 1) + pgpd(xx[xx > u], u, sigmau = 1, xi = 0.5)),
     col = "red", lty = 2, lwd = 2)
lines(xx[xx <= u], kappa * pnorm(xx[xx <= u], 0, 1), col = "blue", lty = 2, lwd = 2)
abline(v = u + epsilon * seq(-1, 1), lty = c(2, 1, 2))
legend('topleft', c('Normal-GPD ITM', 'kappa*Normal', 'kappa*GPD'),
      col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)

# simulated data density histogram and overlay true density 
x = ritmnormgpd(10000, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5)
hist(x, freq = FALSE, breaks = seq(-4, 1000, 0.1), xlim = c(-4, 5))
lines(xx, ditmnormgpd(xx, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5),
  lwd = 2, col = 'black')  
## End(Not run)

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

Holden, L. and Haug, O. (2013). A mixture model for unsupervised tail estimation. arxiv:0902.4137

See Also

normgpd, gpd

and dnorm

Other itmnormgpd: fitmgng, fitmnormgpd, itmgng

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

Author(s)

Alfadino Akbar and Carl Scarrott carl.scarrott@canterbury.ac.nz