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.

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, ...)
``` |

`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. |

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)).

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.

Haizhen Wu and A. Jonathan R. Godfrey.

Updates and bug fixes by Sarah Pirikahu.

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.

ExtDist for other standard distributions.

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.
``` |

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