# Logistic: The Logistic Distribution. In ExtDist: Extending the Range of Functions for Probability Distributions

## 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.

 ``` 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. ```