The Logistic Distribution.

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.

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.

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.

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 = \(\alpha\) and scale = \(\beta\) is most simply defined in terms of its cumulative distribution function (Johnson et.al pp.115-116) $$F(x) = 1- [1 + exp((x-\alpha)/\beta)]^{-1}.$$ The corresponding probability density function is given by $$f(x) = 1/\beta [exp(x-\alpha/\beta][1 + exp(x-\alpha/\beta)]^{-2}$$ Parameter estimation is only implemented numerically.

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

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.

Author

Haizhen Wu and A. Jonathan R. Godfrey.
Updates and bug fixes by Sarah Pirikahu.

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 1.5224838 0.04039734
#>      scale    scale 0.5215655 0.01949557
#> 
#> 
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"]
#> $location
#> [1] 1.522484
#> 
est.par[attributes(est.par)$par.type=="scale"]
#> $scale
#> [1] 0.5215655
#> 

# log-likelihood function
lLogistic(X,param = est.par)
#> [1] -674.8085

# 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 
#> 0.04039734 0.01949557 

# Example of parameter estimation for a distribution with
# unknown parameters currently been sought after.