gngcon function

Normal Bulk with GPD Upper and Lower Tails Extreme Value Mixture Model with Single Continuity Constraint at Thresholds

Density, cumulative distribution function, quantile function and random number generation for the extreme value mixture model with normal for bulk distribution between the upper and lower thresholds with conditional GPD's for the two tails with continuity at the lower and upper thresholds. The parameters are the normal mean nmean and standard deviation nsd, lower tail (threshold ul, GPD shape xil and tail fraction phiul) and upper tail (threshold ur, GPD shape xiR and tail fraction phiuR).

dgngcon(x, nmean = 0, nsd = 1, ul = qnorm(0.1, nmean, nsd),
  xil = 0, phiul = TRUE, ur = qnorm(0.9, nmean, nsd), xir = 0,
  phiur = TRUE, log = FALSE)

pgngcon(q, nmean = 0, nsd = 1, ul = qnorm(0.1, nmean, nsd),
  xil = 0, phiul = TRUE, ur = qnorm(0.9, nmean, nsd), xir = 0,
  phiur = TRUE, lower.tail = TRUE)

qgngcon(p, nmean = 0, nsd = 1, ul = qnorm(0.1, nmean, nsd),
  xil = 0, phiul = TRUE, ur = qnorm(0.9, nmean, nsd), xir = 0,
  phiur = TRUE, lower.tail = TRUE)

rgngcon(n = 1, nmean = 0, nsd = 1, ul = qnorm(0.1, nmean, nsd),
  xil = 0, phiul = TRUE, ur = qnorm(0.9, nmean, nsd), xir = 0,
  phiur = TRUE)

Arguments

• x: quantiles
• nmean: normal mean
• nsd: normal standard deviation (positive)
• ul: lower tail threshold
• xil: lower tail GPD shape parameter
• phiul: probability of being below lower threshold $[0, 1]$ or TRUE
• ur: upper tail threshold
• xir: upper tail GPD shape parameter
• phiur: probability of being above upper 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)

Returns

dgngcon gives the density, pgngcon gives the cumulative distribution function, qgngcon gives the quantile function and rgngcon gives a random sample.

Details

Extreme value mixture model combining normal distribution for the bulk between the lower and upper thresholds and GPD for upper and lower tails with Continuity Constraints at the lower and upper threshold. The user can pre-specify phiul and phiur permitting a parameterised value for the lower and upper tail fraction respectively. Alternatively, when phiul=TRUE or phiur=TRUE the corresponding tail fraction is estimated as from the normal bulk model.

Notice that the tail fraction cannot be 0 or 1, and the sum of upper and lower tail fractions phiul+phiur<1, so the lower threshold must be less than the upper, ul<ur.

The cumulative distribution function now has three components. The lower tail with tail fraction $\phi_{ul}$ defined by the normal bulk model (phiul=TRUE) upto the lower threshold $x < u_l$:

$$F(x) = H(u_l) G_l(x).$$

where $H(x)$ is the normal cumulative distribution function (i.e. pnorm(ur, nmean, nsd)). The $G_l(X)$ is the conditional GPD cumulative distribution function with negated data and threshold, i.e. dgpd(-x, -ul, sigmaul, xil, phiul). The normal bulk model between the thresholds $u_l \le x \le u_r$ given by:

$$F(x) = H(x).$$

Above the threshold $x > u_r$ the usual conditional GPD:

$$F(x) = H(u_r) + [1 - H(u_r)] G(x)$$

where $G(X)$.

The cumulative distribution function for the pre-specified tail fractions $\phi_{ul}$ and $\phi_{ur}$ is more complicated. The unconditional GPD is used for the lower tail $x < u_l$:

$$F(x) = \phi_{ul} G_l(x).$$

The normal bulk model between the thresholds $u_l \le x \le u_r$ given by:

$$F(x) = \phi_{ul}+ (1-\phi_{ul}-\phi_{ur}) (H(x) - H(u_l)) / (H(u_r) - H(u_l)).$$

Above the threshold $x > u_r$ the usual conditional GPD:

$$F(x) = (1-\phi_{ur}) + \phi_{ur} G(x)$$

Notice that these definitions are equivalent when $\phi_{ul} = H(u_l)$ and $\phi_{ur} = 1 - H(u_r)$.

The continuity constraint at ur means that:

$$\phi_{ur} g_r(x) = (1-\phi_{ul}-\phi_{ur}) h(u_r)/ (H(u_r) - H(u_l)).$$

By rearrangement, the GPD scale parameter sigmaur is then:

$$\sigma_ur = \phi_{ur} (H(u_r) - H(u_l))/ h(u_r) (1-\phi_{ul}-\phi_{ur}).$$

where $h(x)$, $g_l(x)$ and $g_r(x)$ are the normal and conditional GPD density functions for lower and upper tail respectively. In the special case of where the tail fraction is defined by the bulk model this reduces to

$$\sigma_ur = [1-H(u_r)] / h(u_r)$$

.

The continuity constraint at ul means that:

$$\phi_{ul} g_l(x) = (1-\phi_{ul}-\phi_{ur}) h(u_l)/ (H(u_r) - H(u_l)).$$

The GPD scale parameter sigmaul is replaced by:

$$\sigma_ul = \phi_{ul} (H(u_r) - H(u_l))/ h(u_l) (1-\phi_{ul}-\phi_{ur}).$$

In the special case of where the tail fraction is defined by the bulk model this reduces to

$$\sigma_ul = H(u_l)/ h(u_l)$$

.

See gpd for details of GPD upper tail component, dnorm for details of normal bulk component, dnormgpd for normal with GPD extreme value mixture model and dgng for normal bulk with GPD upper and lower tails extreme value mixture model.

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 rgngcon 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 rgngcon 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))

x = rgngcon(1000, phiul = 0.15, phiur = 0.15)
xx = seq(-6, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-6, 6))
lines(xx, dgngcon(xx, phiul = 0.15, phiur = 0.15))

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

x = rgngcon(1000, xil = -0.3, phiul = 0.2, xir = 0.3, phiur = 0.2)
xx = seq(-6, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-6, 6))
lines(xx, dgngcon(xx, xil = -0.3, phiul = 0.2, xir = 0.3, phiur = 0.2))

plot(xx, dgngcon(xx, xil = -0.3, phiul = 0.2, xir = 0.3, phiur = 0.2), type = "l", ylim = c(0, 0.4))
lines(xx, dgngcon(xx, xil = -0.3, phiul = 0.3, xir = 0.3, phiur = 0.3), col = "red")
lines(xx, dgngcon(xx, xil = -0.3, phiul = TRUE, xir = 0.3, phiur = TRUE), col = "blue")
legend("topleft", c("phiul = phiur = 0.2", "phiul = phiur = 0.3", "Bulk Tail Fraction"),
  col=c("black", "red", "blue"), lty = 1)
## 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

Zhao, X., Scarrott, C.J. Reale, M. and Oxley, L. (2010). Extreme value modelling for forecasting the market crisis. Applied Financial Econometrics 20(1), 63-72.

See Also

gpd and dnorm

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

Other normgpdcon: fgngcon, fhpdcon, flognormgpdcon, fnormgpdcon, fnormgpd, gng, hpdcon, hpd, normgpdcon, normgpd

Other gng: fgngcon, fgng, fitmgng, fnormgpd, gng, itmgng, normgpd

Other gngcon: fgngcon, fgng, fnormgpdcon, gng, normgpdcon

Other fgngcon: fgngcon

Author(s)

Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz