JohnsonSU {ExtDist}

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.

...

Additional 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.

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.

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.

Author(s)

Haizhen Wu and A. Jonathan R. Godfrey.
Updates and bug fixes by Sarah Pirikahu.

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.

See Also

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