Last data update: 2014.03.03

R: The Logistic Distribution.
 Logistic R Documentation

## 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

```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.

ExtDist for other standard distributions.

### Examples

```# 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.
```

### Results

```
R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(ExtDist)

Attaching package: 'ExtDist'

The following object is masked from 'package:stats':

BIC

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/ExtDist/Logistic.Rd_%03d_medium.png", width=480, height=480)
> ### Name: Logistic
> ### Title: The Logistic Distribution.
> ### Aliases: Logistic dLogistic eLogistic lLogistic pLogistic qLogistic
> ###   rLogistic
>
> ### ** Examples
>
> # 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

Parameters for the Logistic distribution.
(found using the  numerical.MLE method.)

Parameter     Type  Estimate       S.E.
location location 1.5025650 0.03819408
scale    scale 0.4912822 0.01831383

> 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"]
\$location
 1.502565

> est.par[attributes(est.par)\$par.type=="scale"]
\$scale
 0.4912822

>
> # log-likelihood function
> lLogistic(X,param = est.par)
 -642.0165
>
> # 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
location      scale
0.03819408 0.01831383
>
> # Example of parameter estimation for a distribution with
> # unknown parameters currently been sought after.
>
>
>
>
>
> dev.off()
null device
1
>

```