kiener2: Asymmetric Kiener Distribution K2

Description

Density, distribution function, quantile function, random generation, value-at-risk, expected shortfall (+ signed left/right tail mean) and additional formulae for asymmetric Kiener distribution K2.

Usage

dkiener2(x, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE)
pkiener2(q, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE)
qkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE)
rkiener2(n, m = 0, g = 1, a = 3.2, w = 3.2)
dpkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE)
dqkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE)
lkiener2(x, m = 0, g = 1, a = 3.2, w = 3.2)
dlkiener2(lp, m = 0, g = 1, a = 3.2, w = 3.2, log = FALSE)
qlkiener2(lp, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE)
varkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE)
ltmkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE)
rtmkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE)
dtmqkiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE)
eskiener2(p, m = 0, g = 1, a = 3.2, w = 3.2, lower.tail = TRUE,
  log.p = FALSE, signedES = FALSE)

Arguments

x: vector of quantiles.
m: numeric. The median.
g: numeric. The scale parameter, preferably strictly positive.
a: numeric. The left tail parameter, preferably strictly positive.
w: numeric. The right tail parameter, preferably strictly positive.
log: logical. If TRUE, densities are given in log scale.
q: vector of quantiles.
lower.tail: logical. If TRUE, use p. If FALSE, use 1-p.
log.p: logical. If TRUE, probabilities p are given as log(p).
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.
lp: vector of logit of probabilities.
signedES: logical. FALSE (default) returns positive numbers for left and right tails. TRUE returns negative number (= ltmkiener4) for left tail and positive number (= rtmkiener4) for right tail.

Details

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 distributions K2(m, g, a, w) are distributions with asymmetrical left and right fat tails described by the parameters a (alpha) for the left tail and w (omega) for the right tail. These parameters correspond to the power exponent that appear in Pareto formula and Karamata theorems.

As a and w are highly correlated, the use of Kiener distributions (K3(..., k, d) K4 (K4(..., k, e) is an alternate solution.

m is the median of the distribution. g is the scale parameter and the inverse of the density at the median: $ g = 1 / 8 / f(m) $. As a first estimate, it is approximatively one fourth of the standard deviation $ g \approx \sigma / 4 $ but is independant from it.

The d, p functions have no explicit forms. They are provided here for convenience. They are estimated from a reverse optimization on the quantile function and can be (very) slow, depending the number of points to estimate. We recommand to use the quantile function as far as possible. WARNING: Results may become inconsistent when a or w are smaller than 1. Hopefully, this case seldom happens in finance.

qkiener2 function is defined for p in (0, 1) by: $$ qkiener2(p, m, g, a, w) = m + g * k * (- exp(-logit(p)/a) + exp(logit(p)/w) ) $$ where k is the harmonic mean of the tail parameters a and w calculated by $k = aw2k(a, w)$.

rkiener2 generates n random quantiles.

In addition to the classical d, p, q, r functions, the prefixes dp, dq, l, dl, ql are also provided.

dpkiener2 is the density function calculated from the probability p. It is defined for p in (0, 1) by: $$ dpkiener2(p, m, g, a, w) = p * (1 - p) / k / g / ( exp(-logit(p)/a)/a + exp(logit(p)/w)/w $$

dqkiener2 is the derivate of the quantile function calculated from the probability p. It is defined for p in (0, 1) by: $$ dqkiener2(p, m, g, a, w) = k * g / p / (1 - p) * ( exp(-logit(p)/a)/a + exp(logit(p)/w)/w ) $$

lkiener2 function is estimated from a reverse optimization and can be (very) slow depending the number of points to estimate. Initialization is done by assuming a symmetric distribution lkiener1 around the harmonic mean k, then optimization is performed to take into account the true values a and w. The result can be then compared to the empirical probability logit(p). WARNING: Results may become inconsistent when a or w are smaller than 1. Hopefully, this case seldom happens in finance.

dlkiener2 is the density function calculated from the logit of the probability lp = logit(p). it is defined for lp in (-Inf, +Inf) by: $$ dlkiener2(lp, m, g, a, w) = p * (1 - p) / k / g / ( exp(-lp/a)/a + exp(lp/w)/w ) $$

qlkiener2 is the quantile function calculated from the logit of the probability. It is defined for lp in (-Inf, +Inf) by: $$ qlkiener2(lp, m, g, a, w) = m + g * k * ( - exp(-lp/a) + exp(lp/w) ) $$

varkiener2 designates the Value a-risk and turns negative numbers into positive numbers with the following rule: $$ varkiener2 <- if(p <= 0.5) { - qkiener2 } else { qkiener2 } $$ 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.

ltmkiener2, rtmkiener2 and eskiener2 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 qkiener2 divided by p. Right tail mean is the integrale from p to +Inf of the quantile function qkiener2 divided by 1-p. Expected shortfall turns negative numbers into positive numbers with the following rule: $$ eskiener2 <- if(p <= 0.5) { - ltmkiener2 } else { rtmkiener2 } $$ 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.

dtmqkiener2 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.

References

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

Examples

Run this code


require(graphics)

### Example 1
pp <- c(ppoints(11, a = 1), NA, NaN) ; pp
lp <- logit(pp) ; lp
qkiener2(  p = pp, m = 2, g = 1.5, a = 4, w = 6)
qkiener2(  p = pp, m = 2, g = 1.5, a = 4, w = 6)
qlkiener2(lp = lp, m = 2, g = 1.5, a = 4, w = 6)
dpkiener2( p = pp, m = 2, g = 1.5, a = 4, w = 6)
dlkiener2(lp = lp, m = 2, g = 1.5, a = 4, w = 6)
dqkiener2( p = pp, m = 2, g = 1.5, a = 4, w = 6)


### Example 2
a       <- 6
w       <- 4
set.seed(2014)
mainTC  <- paste("qkiener2(p, m = 0, g = 1, a = ", a, ", w = ", w, ")")
mainsum <- paste("cumulated qkiener2(p, m = 0, g = 1, a = ", a, ", w = ", w, ")")
T       <- 500
C       <- 4
TC      <- qkiener2(p = runif(T*C), m = 0, g = 1, a = a, w = w)
matTC   <- matrix(TC, nrow = T, ncol = C, dimnames = list(1:T, letters[1:C]))
head(matTC)
plot.ts(matTC, main = mainTC)
#
matsum  <- apply(matTC, MARGIN=2, cumsum) 
head(matsum)
plot.ts(matsum, plot.type = "single", main = mainsum)
### End example 2


### Example 3 (four plots: probability, density, logit, logdensity)
x     <- q  <- seq(-15, 15, length.out=101)
w     <- c(0.6, 1, 1.5, 2, 3.2, 10) ; names(w) <- w
olty  <- c(2, 1, 2, 1, 2, 1, 1)
olwd  <- c(1, 1, 2, 2, 3, 3, 2)
ocol  <- c(2, 2, 4, 4, 3, 3, 1)
lleg  <- c("logit(0.999) = 6.9", "logit(0.99)   = 4.6", "logit(0.95)   = 2.9", 
           "logit(0.50)   = 0", "logit(0.05)   = -2.9", "logit(0.01)   = -4.6", 
           "logit(0.001) = -6.9  ")
op    <- par(mfrow=c(2,2), mgp=c(1.5,0.8,0), mar=c(3,3,2,1))

plot(x, plogis(x, scale = 2), type = "n", lwd = 2, ylim = c(0, 1),
     xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "pkiener2(q, m, g, a=2, w=...)")
for (i in 1:length(w)) lines(x, pkiener2(x, a = 2, w = w[i]), 
       lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("topleft", title = expression(omega), legend = c(w), 
       cex = 0.7, inset = 0.02, lty = olty, lwd = olwd, col = ocol )

plot(x, dlogis(x, scale = 2), type = "n", lwd = 2, ylim = c(0, 0.17),
     xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "dkiener2(q, m, g, a=2, w=...)")
for (i in 1:length(w)) lines(x, dkiener2(x, a = 2, w = w[i]), 
       lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("topright", title = expression(omega), legend = c(w), 
       cex = 0.7, inset = 0.02, lty = olty, lwd = olwd, col = ocol )

plot(x, x/2, type = "n", lwd = 1, ylim = c(-7.5, 7.5), yaxt="n", xaxs = "i", 
     yaxs = "i", xlab = "", ylab = "", 
     main = "logit(pkiener2(q, m, g, a=2, w=...))")
axis(2, las=1, at=c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9) )
for (i in 1:length(w)) lines(x, lkiener2(x, a = 2, w = w[i]),  
       lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("topleft", legend = lleg, cex = 0.7, inset = 0.02 )
legend("bottomright", title = expression(omega), legend = c(w), 
       cex = 0.7, inset = 0.02, lty = c(olty), lwd = c(olwd), col = c(ocol) )

plot(x, dlogis(x, scale = 2, log=TRUE), type = "n", lwd = 2, ylim = c(-8, -1.5),
     xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "log(dkiener2(q, m, g, a=2, w=...))")
for (i in 1:length(w)) lines(x, dkiener2(x, a = 2, w = w[i], log=TRUE), 
       lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("bottom", title = expression(omega), legend = c(w), 
       cex = 0.7, inset = 0.02, lty = olty, lwd = olwd, col = ocol )
### End example 3


### Example 4 (four plots: quantile, derivate, density and quantiles from p)
p     <- ppoints(199, a=0)
w     <- c(0.6, 1, 1.5, 2, 3.2, 10) ; names(w) <- w ; w
op    <- par(mfrow=c(2,2), mgp=c(1.5,0.8,0), mar=c(3,3,2,1))

plot(p, qlogis(p, scale = 2), type = "l", lwd = 2, xlim = c(0, 1), 
     ylim = c(-15, 15), xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "qkiener2(p, m, g, a=2, w=...)")
for (i in 1:length(w)) lines(p, qkiener2(p, a = 2, w = w[i]), 
          lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("topleft", title = expression(omega), legend = c(w, "qlogis(x/2)"), 
          inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )

plot(p, 2/p/(1-p), type = "l", lwd = 2, xlim = c(0, 1), ylim = c(0, 100),
     xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "dqkiener2(p, m, g, a=2, w=...)")
for (i in 1:length(w)) lines(p, dqkiener2(p, a = 2, w = w[i]), 
          lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("top", title = expression(omega), legend = c(w, "p*(1-p)/2"), 
          inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )

plot(qlogis(p, scale = 2), p*(1-p)/2, type = "l", lwd = 2, xlim = c(-15, 15), 
     ylim = c(0, 0.18), xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "qkiener2, dpkiener2(p, m, g, a=2, w=...)")
for (i in 1:length(w)) { 
     lines(qkiener2(p, a = 2, w = w[i]), dpkiener2(p, a = 2, w = w[i]),
           lty = olty[i], lwd = olwd[i], col = ocol[i] ) }
legend("topleft", title = expression(omega), legend = c(w, "p*(1-p)/2"), 
          inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )

plot(qlogis(p, scale = 2), p, type = "l", lwd = 2, xlim = c(-15, 15), 
     ylim = c(0, 1), xaxs = "i", yaxs = "i", xlab = "", ylab = "", 
     main = "inverse axis qkiener2(p, m, g, a=2, w=...)")
for (i in 1:length(w)) lines(qkiener2(p, a = 2, w = w[i]), p,
          lty = olty[i], lwd = olwd[i], col = ocol[i] )
legend("topleft", title = expression(omega), legend = c(w, "qlogis(x/2)"), 
          inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 )
### End example 4


### Example 5 (q and VaR, ltm, rtm, and ES)
pp <- c(0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, 
        0.10, 0.20, 0.35, 0.5, 0.65, 0.80, 0.90,
        0.95, 0.975, 0.99, 0.995, 0.9975, 0.999)
m <- -10 ; g <- 1 ; a <- 5 ; w = 3 
k <- aw2k(a, w) ; d <- aw2d(a, w) ; e <- aw2e(a, w)
round(c(m = m, g = g, a = a, k = k, w = w, d = d, e = e), 2) 
plot(qkiener2(pp, m, g, a, w), pp, type = "b")
round(cbind(p = pp, "1-p" = 1-pp,
	q   =   qkiener2(pp, m, g, a, w), 
	ltm = ltmkiener2(pp, m, g, a, w), 
	rtm = rtmkiener2(pp, m, g, a, w), 
	ES  =  eskiener2(pp, m, g, a, w), 
	VaR = varkiener2(pp, m, g, a, w)), 4)
round(kmean(c(m, g, a, w), model = "K2"), 4) # limit value for ltm and rtm
round(cbind(p = pp, "1-p" = 1-pp, 
	q   =   qkiener2(pp, m, g, a, w, lower.tail = FALSE), 
	ltm = ltmkiener2(pp, m, g, a, w, lower.tail = FALSE), 
	rtm = rtmkiener2(pp, m, g, a, w, lower.tail = FALSE), 
	ES  =  eskiener2(pp, m, g, a, w, lower.tail = FALSE), 
	VaR = varkiener2(pp, m, g, a, w, lower.tail = FALSE)), 4)
### End example 5

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