# The Logistic Distribution.

### Description

Density, distribution, and quantile, random number generation,
and parameter estimation functions for the logistic distribution with parameters `location`

and `scale`

.
Parameter estimation can be based on a weighted or unweighted i.i.d. sample and can be carried out numerically.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
dLogistic(x, location = 0, scale = 1, params = list(location = 0, scale =
1), ...)
pLogistic(q, location = 0, scale = 1, params = list(location = 0, scale =
1), ...)
qLogistic(p, location = 0, scale = 1, params = list(location = 0, scale =
1), ...)
rLogistic(n, location = 0, scale = 1, params = list(location = 0, scale =
1), ...)
eLogistic(X, w, method = "numerical.MLE", ...)
lLogistic(X, w, location = 0, scale = 1, params = list(location = 0, scale
= 1), logL = TRUE, ...)
``` |

### Arguments

`x,q` |
A vector of quantiles. |

`location` |
Location parameter. |

`scale` |
Scale parameter. |

`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, lLogistic gives the log-likelihood, otherwise the likelihood is given. |

### Details

If `location`

or `scale`

are omitted, they assume the default values of 0 or 1
respectively.

The `dLogistic()`

, `pLogistic()`

, `qLogistic()`

,and `rLogistic()`

functions serve as wrappers of the
standard `dlogis`

, `plogis`

, `qlogis`

, and
`rlogis`

functions in the stats 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 logistic distribution with `location`

= *α* and `scale`

= *β* is most simply
defined in terms of its cumulative distribution function (Johnson et.al pp.115-116)

*F(x) = 1- [1 + exp((x-α)/β)]^{-1}.*

The corresponding probability density function is given by

*f(x) = 1/β [exp(x-α/β][1 + exp(x-α/β)]^{-2}*

Parameter estimation is only implemented numerically.

The score function and Fishers information are as given by Shi (1995) (See also Kotz & Nadarajah (2000)).

### Value

dLogistic gives the density, pLogistic the distribution function, qLogistic the quantile function, rLogistic generates random deviates, and eLogistic estimates the parameters. lLogistic 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. (1995) Continuous Univariate Distributions, volume 2,
chapter 23. Wiley, New York.

Shi, D. (1995) Fisher information for a multivariate extreme value distribution, Biometrika, vol 82, pp.644-649.

Kotz, S. and Nadarajah (2000) Extreme Value Distributions Theory and Applications, chapter 3, Imperial Collage Press,
Singapore.

### See Also

ExtDist for other standard distributions.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | ```
# Parameter estimation for a distribution with known shape parameters
X <- rLogistic(n=500, location=1.5, scale=0.5)
est.par <- eLogistic(X); est.par
plot(est.par)
# Fitted density curve and histogram
den.x <- seq(min(X),max(X),length=100)
den.y <- dLogistic(den.x,location=est.par$location,scale=est.par$scale)
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 location or scale parameters
est.par[attributes(est.par)$par.type=="location"]
est.par[attributes(est.par)$par.type=="scale"]
# log-likelihood function
lLogistic(X,param = est.par)
# Evaluation of the precision of the parameter estimates by the Hessian matrix
H <- attributes(est.par)$nll.hessian
fisher_info <- solve(H)
var <- sqrt(diag(fisher_info));var
# Example of parameter estimation for a distribution with
# unknown parameters currently been sought after.
``` |