ExtDist (version 0.6-3)

# JohnsonSU: The Johnson SU distribution.

## Description

Density, distribution, quantile, random number generation and parameter estimation functions for the Johnson SU (unbounded support) distribution. Parameter estimation can be based on a weighted or unweighted i.i.d sample and can be carried out numerically.

## Usage

dJohnsonSU(x, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2), ...)

pJohnsonSU(q, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2), ...)

qJohnsonSU(p, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2), ...)

rJohnsonSU(n, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2), ...)

eJohnsonSU(X, w, method = "numerical.MLE", ...)

lJohnsonSU(X, w, gamma = -0.5, delta = 2, xi = -0.5, lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2),
logL = TRUE, ...)

## Arguments

x,q
A vector of quantiles.
gamma,delta
Shape parameters.
xi,lambda
Location-scale parameters.
params
A list that includes all named parameters.
...
p
A vector of probabilities.
n
Number of observations.
X
Sample observations.
w
An optional vector of sample weights.
method
Parameter estimation method.
logL
logical; if TRUE, lJohnsonSU gives the log-likelihood, otherwise the likelihood is given.

## Value

• dJohnsonSU gives the density, pJohnsonSU the distribution function, qJohnsonSU gives the quantile function, rJohnsonSU generates random variables, and eJohnsonSU estimates the parameters. lJohnsonSU provides the log-likelihood function.

## Details

The Johnson system of distributions consists of families of distributions that, through specified transformations, can be reduced to the standard normal random variable. It provides a very flexible system for describing statistical distributions and is defined by $$z = \gamma + \delta f(Y)$$ with $Y = (X-xi)/lambda$. The Johnson SB distribution arises when $f(Y) = archsinh(Y)$, where $-\infty < Y < \infty$. This is the unbounded Johnson family since the range of Y is $(-\infty,\infty)$, Karian & Dudewicz (2011). The JohnsonSU distribution has probability density function $$p_X(x) = \frac{\delta}{\sqrt{2\pi((x-xi)^2 + lambda^2)}}exp[-0.5(\gamma + \delta ln(\frac{x-xi + \sqrt{(x-xi)^2 + lambda^2}}{lambda}))^2].$$ Parameter estimation can only be carried out numerically.

## References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions, volume 1, chapter 12, Wiley, New York. Bowman, K.O., Shenton, L.R. (1983). Johnson's system of distributions. In: Encyclopedia of Statistical Sciences, Volume 4, S. Kotz and N.L. Johnson (eds.), pp. 303-314. John Wiley and Sons, New York. Z. A. Karian and E. J. Dudewicz (2011) Handbook of Fitting Statistical Distributions with R, Chapman & Hall.

ExtDist for other standard distributions.

## Examples

# Parameter estimation for a known distribution
X <- rJohnsonSU(n=500, gamma=-0.5, delta=2, xi=-0.5, lambda=2)
est.par <- eJohnsonSU(X); est.par
plot(est.par)

# Fitted density curve and histogram
den.x <- seq(min(X),max(X),length=100)
den.y <- dJohnsonSU(den.x,params = est.par)
hist(X, breaks=10, probability=TRUE, ylim = c(0,1.2*max(den.y)))
lines(den.x, den.y, col="blue")
lines(density(X), lty=2)

# Extracting shape and boundary parameters
est.par[attributes(est.par)$par.type=="shape"] est.par[attributes(est.par)$par.type=="boundary"]

# Parameter Estimation for a distribution with unknown shape parameters
# Example from Karian, Z.A and Dudewicz, E.J. (2011) p.657.
# Parameter estimates as given by Karian & Dudewicz are:
# gamma =-0.2823, delta=1.0592, xi = -1.4475 and lambda = 4.2592  with log-likelihood = -277.1543
data <- c(1.99, -0.424, 5.61, -3.13, -2.24, -0.14, -3.32, -0.837, -1.98, -0.120,
7.81, -3.13, 1.20, 1.54, -0.594, 1.05, 0.192, -3.83, -0.522, 0.605,
0.427, 0.276, 0.784, -1.30, 0.542, -0.159, -1.66, -2.46, -1.81, -0.412,
-9.67, 6.61, -0.589, -3.42, 0.036, 0.851, -1.34, -1.22, -1.47, -0.592,
-0.311, 3.85, -4.92, -0.112, 4.22, 1.89, -0.382, 1.20, 3.21, -0.648,
-0.523, -0.882, 0.306, -0.882, -0.635, 13.2, 0.463, -2.60, 0.281, 1.00,
-0.336, -1.69, -0.484, -1.68, -0.131, -0.166, -0.266, 0.511, -0.198, 1.55,
-1.03, 2.15, 0.495, 6.37, -0.714, -1.35, -1.55, -4.79, 4.36, -1.53,
-1.51, -0.140, -1.10, -1.87, 0.095, 48.4, -0.998, -4.05, -37.9, -0.368,
5.25, 1.09, 0.274, 0.684, -0.105, 20.3, 0.311, 0.621, 3.28, 1.56)
est.par <- eJohnsonSU(data); est.par
plot(est.par)

# Estimates calculated by eJohnsonSU differ from those given by Karian & Dudewicz (2011).
# However, eJohnsonSU's parameter estimates appear to be an improvement, due to a larger
# log-likelihood of -250.3208 (as given by lJohnsonSU below).

# log-likelihood function
lJohnsonSU(data, param = est.par)

# Evaluation of the precision using the Hessian matrix
H <- attributes(est.par)\$nll.hessian
var <- solve(H)
se <- sqrt(diag(var)); se