JohnsonSB {ExtDist} | R Documentation |

## The Johnson SB distribution.

### Description

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

### Usage

```
dJohnsonSB(
x,
gamma = -0.5,
delta = 2,
xi = -0.5,
lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2),
...
)
dJohnsonSB_ab(
x,
gamma = -0.5,
delta = 2,
a = -0.5,
b = 1.5,
params = list(gamma = -0.5, delta = 2, a = -0.5, b = 1.5),
...
)
pJohnsonSB(
q,
gamma = -0.5,
delta = 2,
xi = -0.5,
lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2),
...
)
qJohnsonSB(
p,
gamma = -0.5,
delta = 2,
xi = -0.5,
lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2),
...
)
rJohnsonSB(
n,
gamma = -0.5,
delta = 2,
xi = -0.5,
lambda = 2,
params = list(gamma = -0.5, delta = 2, xi = -0.5, lambda = 2),
...
)
eJohnsonSB(X, w, method = "numerical.MLE", ...)
lJohnsonSB(
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` , `a` , `b` |
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, it is assumed that the log-likelihood is desired. Set to FALSE if the likelihood is wanted. |

### 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) = ln[Y/(1-Y)]`

, where `0 < Y < 1`

.
This is the bounded Johnson family since the range of Y is `(0,1)`

, Karian & Dudewicz (2011).

The `dJohnsonSB()`

, `pJohnsonSB()`

, `qJohnsonSB()`

,and `rJohnsonSB()`

functions serve as wrappers of the
`dJohnson`

, `pJohnson`

, `qJohnson`

, and
`rJohnson`

functions in the SuppDists package. They allow for the parameters to be declared not only as
individual numerical values, but also as a list so parameter estimation can be carried out.

The JohnsonSB distribution has probability density function

`p_X(x) = \frac{\delta lambda}{\sqrt{2\pi}(x-xi)(1- x + xi)}exp[-0.5(\gamma + \delta ln((x-xi)/(1-x+xi)))^2].`

### Value

dJohnsonSB gives the density, pJohnsonSB the distribution function, qJohnsonSB gives quantile function, rJohnsonSB generates random deviates, and eJohnsonSB estimate the parameters. lJohnsonSB provides the log-likelihood function. The dJohnsonSB_ab provides an alternative parameterisation of the JohnsonSB distribution.

### 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.

Kotz, S. and van Dorp, J. R. (2004). Beyond Beta: Other Continuous
Families of Distributions with Bounded Support and Applications. Appendix B.
World Scientific: Singapore.

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 distribution with known shape parameters
X <- rJohnsonSB(n=500, gamma=-0.5, delta=2, xi=-0.5, lambda=2)
est.par <- eJohnsonSB(X); est.par
plot(est.par)
# Fitted density curve and histogram
den.x <- seq(min(X),max(X),length=100)
den.y <- dJohnsonSB(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))
# Extracting location, scale and shape parameters
est.par[attributes(est.par)$par.type=="location"]
est.par[attributes(est.par)$par.type=="scale"]
est.par[attributes(est.par)$par.type=="shape"]
# Parameter Estimation for a distribution with unknown shape parameters
# Example from Karian, Z.A and Dudewicz, E.J. (2011) p.647.
# Original source of brain scan data Dudewich, E.J et.al (1989).
# Parameter estimates as given by Karian & Dudewicz using moments are:
# gamma =-0.2081, delta=0.9167, xi = 95.1280 and lambda = 21.4607 with log-likelihood = -67.03579
brain <- c(108.7, 107.0, 110.3, 110.0, 113.6, 99.2, 109.8, 104.5, 108.1, 107.2, 112.0, 115.5, 108.4,
107.4, 113.4, 101.2, 98.4, 100.9, 100.0, 107.1, 108.7, 102.5, 103.3)
est.par <- eJohnsonSB(brain); est.par
# Estimates calculated by eJohnsonSB differ from those given by Karian & Dudewicz (2011).
# However, eJohnsonSB's parameter estimates appear to be an improvement, due to a larger
# log-likelihood of -66.35496 (as given by lJohnsonSB below).
# log-likelihood function
lJohnsonSB(brain, param = est.par)
```

*ExtDist*version 0.7-2 Index]