Density, distribution function, quantile function, random generation,
value-at-risk, expected shortfall (+ signed left/right tail mean)
and additional formulae for asymmetric Kiener distribution K7 = K2.
With K7, the vector of parameters is provided as coefk, usually estimated
with paramkienerX (and ~X5,~X7) or regkienerLX$coefk.
Main inputs can be supplied as vector (x,q,p) and matrix (coefk)
and the resulting output is a matrix (useful for simulation).
dkiener7(x, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), log = FALSE)pkiener7(q, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE)
qkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE)
rkiener7(n, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), same_p = FALSE)
dpkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), log = FALSE)
dqkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), log = FALSE)
lkiener7(x, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0))
dlkiener7(lp, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), log = FALSE)
qlkiener7(lp, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE)
varkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE)
ltmkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE)
rtmkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE)
dtmqkiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE)
eskiener7(p, coefk = c(0, 1, 3.2, 3.2, 3.2, 0, 0), lower.tail = TRUE,
log.p = FALSE, signedES = FALSE)
vector of quantiles.
vector of 7 parameters c(m,g,a,k,w,d,e)
or matrix with 7 columns.
logical. If TRUE, densities are given in log scale.
vector of quantiles.
logical. If TRUE, use p. If FALSE, use 1-p.
logical. If TRUE, probabilities p are given as log(p).
vector of probabilities.
integer. Number of observations. If length(n) > 1, the length is taken to be the number required.
logical. If FALSE (default), random probabilies are generated on the fly. If TRUE, the same set of random probabilities is used for each line of coefk (if coefk is a matrix).
vector of logit of probabilities.
logical. FALSE (default) returns positive numbers for
left and right tails. TRUE returns negative number
(= ltmkiener7) for left tail and positive number
(= rtmkiener7) for right tail.
Kiener distributions use the following parameters, some of them being redundant.
See aw2k and pk2pk for the formulas and
the conversion between parameters:
m (mu) is the median of the distribution.
g (gamma) is the scale parameter.
a (alpha) is the left tail parameter.
k (kappa) is the harmonic mean of a and w
and describes a global tail parameter.
w (omega) is the right tail parameter.
d (delta) is the distortion parameter.
e (epsilon) is the eccentricity parameter.
Kiener distribution K7 is designed after kiener2
but uses as input coefk rather than m, g, a
and w.
m is the median of the distribution. g is the scale parameter
and is linked for any value of a and w to the density at the
median through the relation
$$ g * dkiener7(x=m, coefk=coefk) = \frac{\pi}{4\sqrt{3}} \approx 0.453 $$
When a = Inf and w = Inf, g is very close to sd(x).
NOTE: In order to match this standard deviation, the value of g has
been updated from versions < 1.9.0 by a factor
\( \frac{2\pi}{\sqrt{3}}\).
The functions dkiener2347, pkiener2347 and lkiener2347
have no explicit forms. Due to a poor optimization algorithm, their
calculations in versions < 1.9 were unreliable. In versions > 1.9, a much better
algorithm was found and the optimization is conducted in a fast way to avoid
a lengthy optimization. The two extreme elements (minimum, maximum) of the
given x or q arguments are sent to a second order optimizer that
minimize the residual error of the lkiener2347 function and return the
estimated lower and upper logit values. Then a sequence of logit values of
length 51 times the length of x or q is generated between these
lower and upper values and the corresponding quantiles are calculated with
the function qlkiener2347. These 51 times more numerous quantiles are
then compared to the original x or q arguments and the closest
values with their associated logit values are selected. The probabilities are then
calculated with the function invlogit and the densities are calculated
with the function dlkiener2347. The accuracy of this approach depends
on the sparsity of the initial x or q sequences. A 4 digits
accuracy can be expected, enough for most usages.
qkiener7 function is defined for p in (0, 1) by:
$$
qkiener7(p, coefk) = m + \frac{\sqrt{3}}{\pi}*g*k*
\left(-exp\left(-\frac{logit(p)}{a} +\frac{logit(p)}{w}\right)\right)
$$
where k is the harmonic mean of the tail parameters a and w
calculated by \(k = aw2k(a, w)\).
rkiener7 generates n random quantiles.
In addition to the classical d, p, q, r functions, the prefixes dp, dq, l, dl, ql are also provided.
dpkiener7 is the density function calculated from the probability p.
It is defined for p in (0, 1) by:
$$
dpkiener7(p, coefk) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}
\left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)
+\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]^{-1}
$$
dqkiener7 is the derivate of the quantile function calculated from
the probability p. It is defined for p in (0, 1) by:
$$
dqkiener7(p, coefk) = \frac{\sqrt{3}}{\pi}\frac{g}{p(1-p)}\frac{k}{2}
\left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)
+\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]
$$
with a and w extracted from coefk.
dlkiener7 is the density function calculated from the logit of the
probability lp = logit(p) defined in (-Inf, +Inf). The formula is adapted
from distribution K2:
$$
dlkiener7(lp, coefk) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}
\left[ +\frac{1}{a}exp\left(-\frac{lp}{a}\right)
+\frac{1}{w}exp\left( \frac{lp}{w}\right) \right]^{-1}
$$
qlkiener7 is the quantile function calculated from the logit of the
probability. It is defined for lp in (-Inf, +Inf) by:
$$
qlkiener7(lp, coefk) = m + \frac{\sqrt{3}}{\pi}*g*k*
\left(-exp\left(-\frac{lp}{a} +\frac{lp}{w}\right)\right)
$$
varkiener7 designates the Value a-risk and turns negative numbers
into positive numbers with the following rule:
$$
varkiener7 <- if\;(p <= 0.5)\;\; (- qkiener7)\;\; else\;\; (qkiener7)
$$
Usual values in finance are p = 0.01, p = 0.05, p = 0.95 and
p = 0.99. lower.tail = FALSE uses 1-p rather than p.
ltmkiener7, rtmkiener7 and eskiener7 are respectively the
left tail mean, the right tail mean and the expected shortfall of the distribution
(sometimes called average VaR, conditional VaR or tail VaR).
Left tail mean is the integrale from -Inf to p of the quantile function
qkiener7 divided by p.
Right tail mean is the integrale from p to +Inf of the quantile function
qkiener7 divided by 1-p.
Expected shortfall turns negative numbers into positive numbers with the following rule:
$$
eskiener7 <- if\;(p <= 0.5)\;\; (- ltmkiener7)\;\; else\;\; (rtmkiener7)
$$
Usual values in finance are p = 0.01, p = 0.025, p = 0.975 and
p = 0.99. lower.tail = FALSE uses 1-p rather than p.
dtmqkiener7 is the difference between the left tail mean and the quantile
when (p <= 0.5) and the difference between the right tail mean and the quantile
when (p > 0.5). It is in quantile unit and is an indirect measure of the tail curvature.
P. Kiener, Explicit models for bilateral fat-tailed distributions and applications in finance with the package FatTailsR, 8th R/Rmetrics Workshop and Summer School, Paris, 27 June 2014. Download it from: https://www.inmodelia.com/exemples/2014-0627-Rmetrics-Kiener-en.pdf
P. Kiener, Fat tail analysis and package FatTailsR, 9th R/Rmetrics Workshop and Summer School, Zurich, 27 June 2015. Download it from: https://www.inmodelia.com/exemples/2015-0627-Rmetrics-Kiener-en.pdf
C. Acerbi, D. Tasche, Expected shortfall: a natural coherent alternative to Value at Risk, 9 May 2001. Download it from: https://www.bis.org/bcbs/ca/acertasc.pdf
Symmetric Kiener distribution K1 kiener1,
asymmetric Kiener distributions K2, K3 and K4
kiener2, kiener3, kiener4,
conversion functions aw2k,
estimation function paramkienerX,
estimation function fitkienerX,
regression function regkienerLX.
head(ED <- fatreturns(extractData()))
(coefk <- paramkienerX(ED, dgts = 3))
x <- -4
xx <- -4:4
p <- 0.1
pp <- pprobs2
dkiener7(x)
dkiener7(x, coefk)
dkiener7(xx)
dkiener7(xx, coefk)
pkiener7(x)
pkiener7(x, coefk)
pkiener7(xx)
pkiener7(xx, coefk)
qkiener7(p)
qkiener7(p, coefk)
qkiener7(pp)
qkiener7(pp, coefk)
rkiener7(10)
rkiener7(10, coefk)
varkiener7(p)
varkiener7(p, coefk)
varkiener7(pp)
varkiener7(pp, coefk)
ltmkiener7(p)
ltmkiener7(p, coefk)
ltmkiener7(pp)
ltmkiener7(pp, coefk)
eskiener7(p)
eskiener7(p, coefk)
eskiener7(pp)
eskiener7(pp, coefk)
Run the code above in your browser using DataLab