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

Description

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).

Usage

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

quantiles

nmean

normal mean

nsd

normal standard deviation (positive)

lower tail threshold

xil

lower tail GPD shape parameter

phiul

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

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

quantiles

lower.tail

logical, if FALSE then upper tail probabilities

cumulative probabilities

sample size (positive integer)

Value

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 $ϕ_{u l}$ 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 $u_{l} \leq x \leq u_{r}$ 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 $ϕ_{u l}$ and $ϕ_{u r}$ is more complicated. The unconditional GPD is used for the lower tail $x < u_{l}$ : $F (x) = ϕ_{u l} G_{l} (x) .$ The normal bulk model between the thresholds $u_{l} \leq x \leq u_{r}$ given by: $F (x) = ϕ_{u l} + (1 - ϕ_{u l} - ϕ_{u r}) (H (x) - H (u_{l})) / (H (u_{r}) - H (u_{l})) .$ Above the threshold $x > u_{r}$ the usual conditional GPD: $F (x) = (1 - ϕ_{u r}) + ϕ_{u r} G (x)$ Notice that these definitions are equivalent when $ϕ_{u l} = H (u_{l})$ and $ϕ_{u r} = 1 - H (u_{r})$ .

The continuity constraint at ur means that: $ϕ_{u r} g_{r} (x) = (1 - ϕ_{u l} - ϕ_{u r}) h (u_{r}) / (H (u_{r}) - H (u_{l})) .$ By rearrangement, the GPD scale parameter sigmaur is then: $σ_{u} r = ϕ_{u r} (H (u_{r}) - H (u_{l})) / h (u_{r}) (1 - ϕ_{u l} - ϕ_{u r}) .$ 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 $σ_{u} r = [1 - H (u_{r})] / h (u_{r})$ .

The continuity constraint at ul means that: $ϕ_{u l} g_{l} (x) = (1 - ϕ_{u l} - ϕ_{u r}) h (u_{l}) / (H (u_{r}) - H (u_{l})) .$ The GPD scale parameter sigmaul is replaced by: $σ_{u} l = ϕ_{u l} (H (u_{r}) - H (u_{l})) / h (u_{l}) (1 - ϕ_{u l} - ϕ_{u r}) .$ In the special case of where the tail fraction is defined by the bulk model this reduces to $σ_{u} l = H (u_{l}) / h (u_{l})$ .

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.

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.

Examples

Run this code

# 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)
# }
# NOT RUN {
# }

Run the code above in your browser using DataLab

Last chance! 50% off unlimited learning