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)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
$\phi_{ur} = 1 - H(u_r)$.
The continuity constraint at the 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 $$\sigma_ul = H(u_l)/ h(u_l)$$.
The continuity Constraint at the ur means that:
$$\phi_{ur} g_r(x) = (1-\phi_{ul}-\phi_{ur}) h(u_l)/
(H(u_r) - H(u_l)).$$ By rearrangement, the GPD scale
parameter sigmaur is then: $$\sigma_ur =
\phi_{ur} (H(u_r) - H(u_l))/ h(u_l)
(1-\phi_{ul}-\phi_{ur}).$$ where $h(x)$, $g_l(x)$
and $g_r(x)$ are the normal and conditional GPD
density functions for lower and upper tail (i.e.
dnorm(x, nmean, nsd), dgpd(-x, -ul, sigmaul,
xil, phiul), and dgpd(x, ur, sigmaur, xir,
phiur)) In the special case of where the tail fraction
is defined by the bulk model this reduces to
$$\sigma_ur = [1-H(u_r)] / h(u_r)$$.
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. gng,
normgpd,
gpd and
dnorm
Other gngcon: fgngcon,
lgngcon, nlgngconpar(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)Run the code above in your browser using DataLab