evmix (version 1.0)

gng: Normal Bulk with GPD Upper and Lower Tails 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 between the upper and lower thresholds with conditional GPD's for the two tails. The parameters are the normal mean nmean and standard deviation nsd, lower tail (threshold ul, GPD scale sigmaul and shape xil and tail fraction phiul) and upper tail (threshold ur, GPD scale sigmaur and shape xiR and tail fraction phiuR).

Usage

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

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

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

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

Arguments

ul

lower tail threshold

sigmaul

lower tail GPD scale parameter (non-negative)

xil

lower tail GPD shape parameter

phiul

probability of being below lower threshold (0,1)

ur

upper tail threshold

sigmaur

upper tail GPD scale parameter (non-negative)

xir

upper tail GPD shape parameter

phiur

probability of being above upper threshold (0,1)

x

quantile

nmean

normal mean

nsd

normal standard deviation (non-negative)

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

• dgng gives the density, pgng gives the cumulative distribution function, qgng gives the quantile function and rgng 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. 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< code="">, so the lower threshold must be less than the upper, ul. 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 $\phi_{ur} = 1 - H(u_r)$. See gpd for details of GPD upper tail component, dnorm for details of normal bulk component and dnormgpd for normal with GPD 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.

See Also

normgpd, gpd and dnorm Other gng: fgng, lgng, nlgng

Examples

par(mfrow=c(2,2))
x = rgng(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, dgng(xx, phiul = 0.15, phiur = 0.15))

# three tail behaviours
plot(xx, pgng(xx), type = "l")
lines(xx, pgng(xx, xil = 0.3, xir = 0.3), col = "red")
lines(xx, pgng(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 = rgng(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, dgng(xx, xil = -0.3, phiul = 0.2, xir = 0.3, phiur = 0.2))

plot(xx, dgng(xx, xil = -0.3, phiul = 0.2, xir = 0.3, phiur = 0.2), type = "l", ylim = c(0, 0.4))
lines(xx, dgng(xx, xil = -0.3, phiul = 0.3, xir = 0.3, phiur = 0.3), col = "red")
lines(xx, dgng(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)