# Symmetric Kiener Distribution K1 Density, distribution function, quantile function, random generation, value-at-risk, expected shortfall (+ signed left/right tail mean) and additional formulae for symmetric Kiener distribution K1. This distribution is similar to the power hyperbola logistic distribution but with additional parameters for location (`m`) and scale (`g`). ```r dkiener1(x, m = 0, g = 1, k = 3.2, log = FALSE) pkiener1(q, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE) qkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE) rkiener1(n, m = 0, g = 1, k = 3.2) dpkiener1(p, m = 0, g = 1, k = 3.2, log = FALSE) dqkiener1(p, m = 0, g = 1, k = 3.2, log = FALSE) lkiener1(x, m = 0, g = 1, k = 3.2) dlkiener1(lp, m = 0, g = 1, k = 3.2, log = FALSE) qlkiener1(lp, m = 0, g = 1, k = 3.2, lower.tail = TRUE) varkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE) ltmkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE) rtmkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE) dtmqkiener1(p, m = 0, g = 1, k = 3.2, lower.tail = TRUE, log.p = FALSE) eskiener1(p, m = 0, g = 1, k = 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. - `k`: numeric. The 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 (= `ltmkiener1`) for left tail and positive number (= `rtmkiener1`) 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 `K1(m, g, k, ...)` describe distributions with symmetric left and right fat tails with tail parameter `k`. This parameter is the power exponent mentionned in Pareto formula and Karamata theorems. `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. `dkiener1` function is defined for x in (-Inf, +Inf) by: $$ dkiener1(x, m, g, k) =1 / 4 / g / cosh( ashp((x - m)/g, k) )/ (1 + cosh( kashp((x - m)/g, k))) $$ `pkiener1` function is defined for q in (-Inf, +Inf) by: $$ pkiener1(q, m, g, k) = 1/(1 + exp(- kashp((q - m)/g, k))) $$ `qkiener1` function is defined for p in (0, 1) by: $$ qkiener1(p, m, g, k) = m + 2 * g * k * sinh( logit(p)/k ) $$ `rkiener1` generates `n` random quantiles. In addition to the classical d, p, q, r functions, the prefixes dp, dq, l, dl, ql are also provided. `dpkiener1` is the density function calculated from the probability p. It is defined for p in (0, 1) by: $$ dpkiener1(p, m, g, k) = p * (1 - p) / 2 / g / cosh( logit(p)/k ) $$ `dqkiener1` is the derivate of the quantile function calculated from the probability p. It is defined for p in (0, 1) by: $$ dqkiener1(p, m, g, k) = 2 * g / p / (1 - p) * cosh( logit(p)/k ) $$ `lkiener1` function is equivalent to kashp function but with additional parameters `m` and `g`. Being computed from the x (or q) vector, it can be compared to the logit of the empirical probability logit(p) through a nonlinear regression with ordinary or weighted least squares to estimate the distribution parameters. It is defined for x in (-Inf, +Inf) by: $$ lkiener1(x, m, g, k) = kashp((x - m)/g, k) $$ `dlkiener1` is the density function calculated from the logit of the probability lp = logit(p). It is defined for lp in (-Inf, +Inf) by: $$ dlkiener1(lp, m, g, k) = p * (1 - p) / 2 / g / cosh( lp/k ) $$ `qlkiener1` is the quantile function calculated from the logit of the probability lp = logit(p). It is defined for lp in (-Inf, +Inf) by: $$ qlkiener1(lp, m, g, k) = m + g * k * 2 * sinh( lp/k ) $$ `varkiener1` designates the Value a-risk and turns negative numbers into positive numbers with the following rule: $$ varkiener1 <- if(p <= 0.5) (- qkiener1) else (qkiener1) $$ 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`. `ltmkiener1`, `rtmkiener1` and `eskiener1` 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 `qkiener1` divided by `p`. Right tail mean is the integrale from `p` to `+Inf` of the quantile function `qkiener1` divided by 1-p. Expected shortfall turns negative numbers into positive numbers with the following rule: $$ eskiener1 <- if(p <= 0.5) (- ltmkiener1) else (rtmkiener1) $$ 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`. `dtmqkiener1` 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 pp <- c(ppoints(11, a = 1), NA, NaN) ; pp qkiener1(p = pp, k = 4) ### Example 2: Try various value of k = 1.5, 3, 5, 10 k <- 5 # 1.5, 3, 5, 10 set.seed(2014) mainTC <- paste("qkiener1(p, m = 0, g = 1, k = ", k, ")") mainsum <- paste("cumulated qkiener1(p, m = 0, g = 1, k = ", k, ")") T <- 500 C <- 4 TC <- qkiener1(p = runif(T*C), m = 0, g = 1, k = k) 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) k <- c(0.6, 1, 1.5, 2, 3.2, 10) ; names(k) <- k ; k 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 = "b", lwd = 2, ylim = c(0, 1), xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "pkiener1(q, m, g, k)") for (i in 1:length(k)) lines(x, pkiener1(x, k = k[i]), lty = olty[i], lwd = olwd[i], col = ocol[i] ) legend("topleft", title = expression(kappa), legend = c(k, "logistic"), cex = 0.7, inset = 0.02, lty = olty, lwd = olwd, col = ocol ) plot(x, dlogis(x, scale = 2), type = "b", lwd = 2, ylim = c(0, 0.14), xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "dkiener1(x, m, g, k)" ) for (i in 1:length(k)) lines(x, dkiener1(x, k = k[i]), lty = olty[i], lwd = olwd[i], col = ocol[i] ) legend("topright", title = expression(kappa), legend = c(k, "logistic"), cex = 0.7, inset = 0.02, lty = olty, lwd = olwd, col = ocol ) plot(x, x/2, type = "b", lwd = 2, ylim = c(-7.5, 7.5), yaxt="n", xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "logit(pkiener1(q, m, g, k))") axis(2, las=1, at=c(-6.9, -4.6, -2.9, 0, 2.9, 4.6, 6.9) ) for (i in 1:length(k)) lines(x, lkiener1(x, k = k[i]), lty = olty[i], lwd = olwd[i], col = ocol[i] ) lines(x, logit(pnorm(x, 0, 3.192)), type="l", lty=1, lwd=3, col=7) # erfx legend("topleft", legend = lleg, cex = 0.7, inset = 0.02 ) legend("bottomright", title = expression(kappa), legend = c(k, "logistic", "Gauss"), cex = 0.7, inset = 0.02, lty = c(olty, 1), lwd = c(olwd, 3), col = c(ocol , 7) ) plot(x, log(dlogis(x, scale = 2)), lwd = 2, type = "b", ylim = c(-8, -1.5), xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "log(dkiener1(x, m, g, k))") for (i in 1:length(k)) lines(x, log(dkiener1(x, k = k[i])), lty = olty[i], lwd = olwd[i], col = ocol[i] ) lines(x, dnorm(x, 0, 3.192, log = TRUE), type = "l", lty = 1, lwd = 3, col = 7) legend("bottom", title = expression(kappa), legend = c(k, "logistic", "Gauss"), cex = 0.7, inset = 0.02, lty = c(olty, 1), lwd = c(olwd, 3), col = c(ocol , 7) ) ### End example 3 ### Example 4 (four plots: quantile, derivate, density and quantiles from p) p <- ppoints(199, a=0) k <- c(0.6, 1, 1.5, 2, 3.2, 10) ; names(k) <- k ; k 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 = "o", lwd = 2, ylim = c(-15, 15), xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "qkiener1(p, m, g, k)") for (i in 1:length(k)) lines(p, qkiener1(p, k = k[i]), lty = olty[i], lwd = olwd[i], col = ocol[i] ) legend("topleft", title = expression(kappa), legend = c(k, "qlogis(x/2)"), inset = 0.02, lty = olty, lwd = olwd, col = ocol, cex = 0.7 ) plot(p, 2/p/(1-p), type = "o", lwd = 2, xlim = c(0, 1), ylim = c(0, 100), xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "dqkiener1(p, m, g, k)") for (i in 1:length(k)) lines(p, dqkiener1(p, k = k[i]), lty = olty[i], lwd = olwd[i], col = ocol[i] ) legend("top", title = expression(kappa), legend = c(k, "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 = "o", lwd = 2, xlim = c(-15, 15), ylim = c(0, 0.14), xaxs = "i", yaxs = "i", xlab = "", ylab = "", main = "qkiener1, dpkiener1(p, m, g, k)") for (i in 1:length(k)) lines(qkiener1(p, k = k[i]), dpkiener1(p, k = k[i]), lty = olty[i], lwd = olwd[i], col = ocol[i] ) legend("topleft", title = expression(kappa), legend = c(k, "p*(1-p)/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 ; k <- 4 round(c(m = m, g = g, a = k, k = k, w = k, d = 0, e = 0), 2) plot(qkiener1(pp, m, g, k), pp, type = "b") round(cbind(p = pp, "1-p" = 1-pp, q = qkiener1(pp, m, g, k), ltm = ltmkiener1(pp, m, g, k), rtm = rtmkiener1(pp, m, g, k), es = eskiener1(pp, m, g, k), VaR = varkiener1(pp, m, g, k)), 4) round(kmean(c(m, g, k), model = "K1"), 4) # limit value of ltm, rtm round(cbind(p = pp, "1-p" = 1-pp, q = qkiener1(pp, m, g, k, lower.tail = FALSE), ltm = ltmkiener1(pp, m, g, k, lower.tail = FALSE), rtm = rtmkiener1(pp, m, g, k, lower.tail = FALSE), es = eskiener1(pp, m, g, k, lower.tail = FALSE), VaR = varkiener1(pp, m, g, k, lower.tail = FALSE)), 4) ### End example 5 ``` ## 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 Power hyperbola logistic distribution `logishp`, asymmetric Kiener distributions K2, K3, K4 and K7 `kiener2`, `kiener3`, `kiener4`, `kiener7`, regression function `regkienerLX`.

kiener1 function