# Asymmetric Kiener Distribution K4 Density, distribution function, quantile function, random generation, value-at-risk, expected shortfall (+ signed left/right tail mean) and additional formulae for asymmetric Kiener distribution K4. ```r dkiener4(x, m = 0, g = 1, k = 3.2, e = 0, log = FALSE) pkiener4(q, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE, log.p = FALSE) qkiener4(p, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE, log.p = FALSE) rkiener4(n, m = 0, g = 1, k = 3.2, e = 0) dpkiener4(p, m = 0, g = 1, k = 3.2, e = 0, log = FALSE) dqkiener4(p, m = 0, g = 1, k = 3.2, e = 0, log = FALSE) lkiener4(x, m = 0, g = 1, k = 3.2, e = 0) dlkiener4(lp, m = 0, g = 1, k = 3.2, e = 0, log = FALSE) qlkiener4(lp, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE) varkiener4(p, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE, log.p = FALSE) ltmkiener4(p, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE, log.p = FALSE) rtmkiener4(p, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE, log.p = FALSE) dtmqkiener4(p, m = 0, g = 1, k = 3.2, e = 0, lower.tail = TRUE, log.p = FALSE) eskiener4(p, m = 0, g = 1, k = 3.2, e = 0, 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. - `k`: numeric. The tail parameter, preferably strictly positive. - `e`: numeric. The eccentricity parameter between left and right tails. - `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 `K4(m, g, k, e, ...)` are distributions with asymmetrical left and right fat tails described by a global tail parameter `k` and an eccentricity parameter `e`. Distributions K3 (`kiener3`) with parameters `k` (kappa) and `d` (delta) and distributions K4 (`kiener4`) with parameters `k` (kappa) and `e` (epsilon)) have been created to disantangle the parameters `a` (alpha) and `w` (omega) of distributions K2 (`kiener2`). The tiny difference between distributions K3 and K4 ($d = e/k$) has not yet been fully evaluated. Both should be tested at that moment. `k` is the harmonic mean of `a` and `w` and represents a global tail parameter. `e` is an eccentricity parameter between the left tail parameter `a` and the right tail parameter `w`. It verifies the inequality: $-1 < e < 1$ (whereas `d` of distribution K3 verifies $-k < d < k$). The conversion functions (see `aw2k`) are: $$ 1/k = (1/a + 1/w)/2 $$ $$ e = (a - w)/(a + w) $$ $$ a = k/(1 - e) $$ $$ w = k/(1 + e) $$ `e` (and `d`) should be of the same sign than the skewness. A negative value $ e < 0 $ implies $ a < w $ and indicates a left tail heavier than the right tail. A positive value $ e > 0 $ implies $ a > w $ and a right tail heavier than the left tail. `m` is the median of the distribution. `g` is the scale parameter and is linked for any value of `k` and `e` to the density at the median through the relation $$ g * dkiener4(x=m, g=g, e=e) = \frac{\pi}{4\sqrt{3}} \approx 0.453%g * dkiener4(x=m, g=g, e=e) = pi/4/sqrt(3) = 0.453 approximatively $$ When `k = 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 $ 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. `qkiener4` function is defined for p in (0, 1) by: $$ qkiener4(p,m,g,k,e) = m + \frac{\sqrt{3}}{\pi}*g*k*\sinh\left(\frac{logit(p)}{k}\right)*exp\left(\frac{e}{k} logit(p)\right)%qkiener4(p,m,g,k,e) = m + sqrt(3)/pi*g*k*sinh(logit(p)/k)*exp(e/k*logit(p)) $$ `rkiener4` generates `n` random quantiles. In addition to the classical d, p, q, r functions, the prefixes dp, dq, l, dl, ql are also provided. `dpkiener4` is the density function calculated from the probability p. The formula is adapted from distribution K2. It is defined for p in (0, 1) by: $$ dpkiener4(p,m,g,k,e) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}\left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)+\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]^{-1}%dpkiener4(p,m,g,k,e) = pi/sqrt(3)*p*(1-p)/g*2/k/(+exp(-logit(p)/a)/a +exp(logit(p)/w)/w) $$ with `a` and `w` defined from `k` and `e`. `dqkiener4` is the derivate of the quantile function calculated from the probability p. The formula is adapted from distribution K2. It is defined for p in (0, 1) by: $$ dqkiener4(p,m,g,k,e) = \frac{\sqrt{3}}{\pi}\frac{g}{p(1-p)}\frac{k}{2}\left[ +\frac{1}{a}exp\left(-\frac{logit(p)}{a}\right)+\frac{1}{w}exp\left( \frac{logit(p)}{w}\right) \right]%dqkiener4(p,m,g,k,e) = sqrt(3)/pi*g/p/(1-p)*k/2*( exp(-logit(p)/a)/a + exp(logit(p)/w)/w) $$ with `a` and `w` defined with the formula presented above. `dlkiener4` 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: $$ dlkiener2(lp,m,g,k,e) = \frac{\pi}{\sqrt{3}}\frac{p(1-p)}{g}\frac{2}{k}\left[ +\frac{1}{a}exp\left(-\frac{lp}{a}\right)+\frac{1}{w}exp\left( \frac{lp}{w}\right) \right]^{-1}%dlkiener2(lp,m,g,k,e) = pi/sqrt(3)*p*(1-p)/g*2/k/(+exp(-lp/a)/a +exp(lp/w)/w) $$ with `a` and `w` defined above. `qlkiener4` is the quantile function calculated from the logit of the probability. It is defined for lp in (-Inf, +Inf) by: $$ qlkiener4(lp,m,g,k,e) = m + \frac{\sqrt{3}}{\pi}*g*k*\sinh\left(\frac{lp}{k}\right)*exp\left(\frac{e}{k} lp\right)%qlkiener4(lp,m,g,k,e) = m + sqrt(3)/pi*g*k*sinh(lp/k)*exp(e/k*lp) $$ `varkiener4` designates the Value a-risk and turns negative numbers into positive numbers with the following rule: $$ varkiener4 <- if\;(p <= 0.5)\;\; (- qkiener4)\;\; else\;\; (qkiener4)%varkiener4 <- if (p <= 0.5) (- qkiener4) else (qkiener4) $$ 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`. `ltmkiener4`, `rtmkiener4` and `eskiener4` 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 `qkiener4` divided by `p`. Right tail mean is the integrale from `p` to `+Inf` of the quantile function `qkiener4` divided by 1-p. Expected shortfall turns negative numbers into positive numbers with the following rule: $$ eskiener4 <- if\;(p <= 0.5)\;\; (- ltmkiener4)\;\; else\;\; (rtmkiener4)%eskiener4 <- if(p <= 0.5) (- ltmkiener4) else (rtmkiener4) $$ 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`. `dtmqkiener4` 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. ## Examples ```r require(graphics) ### EXAMPLE 1 x <- seq(-5, 5, by = 0.1) ; round(x, 2) round(pkiener4(x, m=0, g=1, k=4, e=0.1), 4) round(dkiener4(x, m=0, g=1, k=4, e=0.1), 4) round(lkiener4(x, m=0, g=1, k=4, e=0.1), 4) plot( x, pkiener4(x, m=0, g=1, k=9999, e=0), las=1, type="l", lwd=2) lines(x, pkiener4(x, m=0, g=1, k=4, e=0.5), col="red") lines(x, pkiener4(x, m=0, g=1, k=4, e=1), lwd=1) # e in [-1, 1] plot( x, dkiener4(x, m=0, g=1, k=9999, e=0), las=1, type="l", lwd=2, ylim=c(0,0.6)) lines(x, dkiener4(x, m=0, g=1, k=4, e=0.5), col="red") lines(x, dkiener4(x, m=0, g=1, k=4, e=1), lwd=1) plot( x, lkiener4(x, m=0, g=1, k=9999, e=0), las=1, type="l", lwd=2) lines(x, lkiener4(x, m=0, g=1, k=4, e=0.05), col="green") lines(x, lkiener4(x, m=0, g=1, k=4, e=0.5), col="red") lines(x, lkiener4(x, m=0, g=1, k=4, e=1), lwd=1) p <- c(ppoints(11, a = 1), NA, NaN) ; p qkiener4(p, k=4, e=0.5) dpkiener4(p, k=4, e=0.5) dqkiener4(p, k=4, e=0.5) varkiener4(p=0.01, k=4, e=0.5) ltmkiener4(p=0.01, k=4, e=0.5) eskiener4(p=0.01, k=4, e=0.5) # VaR and ES should be positive ### END EXAMPLE 1 ### PREPARE THE GRAPHICS FOR EXAMPLES 2 AND 3 xx <- c(-4,-2, 0, 2, 4) lty <- c( 3, 2, 1, 4, 5, 1) lwd <- c( 1, 1, 2, 1, 1, 1) col <- c("cyan3","green3","black","dodgerblue2","purple2","brown3") lat <- c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9) lgt <- 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 ") funleg <- function(xy, e) legend(xy, title = expression(epsilon), legend = names(e), lty = lty, col = col, lwd = lwd, inset = 0.02, cex = 0.8) funlgt <- function(xy) legend(xy, title = "logit(p)", legend = lgt, inset = 0.02, cex = 0.6) ### EXAMPLE 2 ### PROBA, DENSITY, LOGIT-PROBA, LOG-DENSITY FROM x x <- seq(-5, 5, by = 0.1) ; head(x, 10) e <- c(-0.5, -0.25, 0, 0.25, 0.50, 1) ; names(e) <- e fun1 <- function(e, x) pkiener4(x, k=4, e=e) fun2 <- function(e, x) dkiener4(x, k=4, e=e) fun3 <- function(e, x) lkiener4(x, k=4, e=e) fun4 <- function(e, x) dkiener4(x, k=4, e=e, log=TRUE) mat11 <- sapply(e, fun1, x) ; head(mat11, 10) mat12 <- sapply(e, fun2, x) ; head(mat12, 10) mat13 <- sapply(e, fun3, x) ; head(mat13, 10) mat14 <- sapply(e, fun4, x) ; head(mat14, 10) op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1) matplot(x, mat11, type="l", lwd=lwd, lty=lty, col=col, main="pkiener4(x, m=0, g=1, k=4, e=e)", xlab="", ylab="") funleg("topleft", e) matplot(x, mat12, type="l", lwd=lwd, lty=lty, col=col, main="dkiener4", xlab="", ylab="") funleg("topleft", e) matplot(x, mat13, type="l", lwd=lwd, lty=lty, col=col, yaxt="n", ylim=c(-9,9), main="lkiener4", xlab="", ylab="") axis(2, at=lat, las=1) funleg("bottomright", e) funlgt("topleft") matplot(x, mat14, type="l", lwd=lwd, lty=lty, col=col, ylim=c(-8,0), main="log(dkiener4)", xlab="", ylab="") funleg("bottom", e) par(op) ### END EXAMPLE 2 ### EXAMPLE 3 ### QUANTILE, DIFF-QUANTILE, DENSITY, LOG-DENSITY FROM p p <- ppoints(1999, a=0) ; head(p, n=10) e <- c(-0.5, -0.25, 0, 0.25, 0.50, 1) ; names(e) <- e mat15 <- outer(p, e, \(p,e) qkiener4(p, k=4, e=e)) ; head(mat15, 10) mat16 <- outer(p, e, \(p,e) dqkiener4(p, k=4, e=e)) ; head(mat16, 10) mat17 <- outer(p, e, \(p,e) dpkiener4(p, k=4, e=e)) ; head(mat17, 10) op <- par(mfcol = c(2,2), mar = c(2.5,3,1.5,1), las=1) matplot(p, mat15, type="l", xlim=c(0,1), ylim=c(-5,5), lwd=lwd, lty=lty, col=col, las=1, main="qkiener4(p, m=0, g=1, k=4, e=e)", xlab="", ylab="") funleg("topleft", e) matplot(p, mat16, type="l", xlim=c(0,1), ylim=c(0,40), lwd=lwd, lty=lty, col=col, las=1, main="dqkiener4", xlab="", ylab="") funleg("top", e) plot(NA, NA, xlim=c(-5, 5), ylim=c(0, 0.6), las=1, main="qkiener4, dpkiener4", xlab="", ylab="") invisible(mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(mat17), lwd=lwd, lty=1, col=col)) funleg("topright", e) plot(NA, NA, xlim=c(-5, 5), ylim=c(-7, -0.5), las=1, main="qkiener4, log(dpkiener4)", xlab="", ylab="") invisible(mapply(matlines, x=as.data.frame(mat15), y=as.data.frame(log(mat17)), lwd=lwd, lty=lty, col=col)) funleg("bottom", e) par(op) ### END EXAMPLE 3 ### EXAMPLE 4: PROCESSUS: which processus look credible? ### PARAMETER e VARIES, k=4 IS CONSTANT ### RUN SEED ii <- 1 THEN THE cairo_pdf CODE WITH THE 6 SEEDS # cairo_pdf("K4-6x6-stocks-e.pdf") # for (ii in c(1,2016,2018,2022,2023,2024)) { ii <- 1 set.seed(ii) p <- sample(ppoints(299, a=0), 299) e <- c(-0.1, -0.05, 0, 0.05, 0.1, 0.25) ; names(e) <- e mat18 <- outer(p, e, \(p,e) qkiener4(p=p, g=0.85, k=4, e=e)) mat19 <- apply(mat18, 2, cumsum) title <- paste0( "stock_", ii, ": k = 4", ", e_left = c(", paste(e[1:3], collapse = ", "), ")", ", e_right = c(", paste(e[4:6], collapse = ", "), ")") plot.ts(mat19, ann=FALSE, las=1, mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5)) mtext(title, outer = TRUE, line=-1.5, font=2) plot.ts(mat18, ann=FALSE, las=1, mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5)) mtext(title, outer=TRUE, line=-1.5, font=2) # } # dev.off() ### PARAMETER k VARIES, e=0.05 IS CONSTANT # cairo_pdf("K4-6x6-stocks-k.pdf", width=11) # for (ii in c(1,2016,2018,2022,2023,2024)) { ii <- 1 set.seed(ii) p <- sample(ppoints(299, a=0), 299) k <- c(9999, 6, 4, 3, 2, 1) ; names(k) <- k mat20 <- outer(p, k, \(p,k) qkiener4(p=p, g=0.85, k=k, e=0.05)) mat21 <- apply(mat20, 2, cumsum) title <- paste0( "stock_", ii, ": k_left = c(", paste(k[1:3], collapse = ", "), ")", ", k_right = c(", paste(k[4:6], collapse = ", "), ")", ", e = 0.05") plot.ts(mat21, ann=FALSE, las=1, nc=2, mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5)) mtext(title, outer = TRUE, line=-1.5, font=2) plot.ts(mat20, ann=FALSE, las=1, nc=2, mar.multi=c(0,3,0,1), oma.multi=c(3,0,3,0.5)) mtext(title, outer=TRUE, line=-1.5, font=2) # } # dev.off() ### END EXAMPLE 4 ``` ## 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](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](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](https://www.bis.org/bcbs/ca/acertasc.pdf) ## See Also Symmetric Kiener distribution K1 `kiener1`, asymmetric Kiener distributions K2, K3 and K7 `kiener2`, `kiener3`, `kiener7`, conversion functions `aw2k`, estimation function `fitkienerX`,

kiener4 function