Logistic-uniform distribution

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In probability and statistics, the logistic-uniform (LU) distribution is a parametric family of probability distributions with support on the unit interval. It is similar to the beta distribution, but its cumulative distribution function, probability density function, and quantile function are all expressible in terms of elementary functions, and it is easier to sample from. This distribution was described and analyzed by. It was also described by under the name sigmoid-logit distribution. It is part of the logistic-X family of distributions.

Description

The parameters of the distribution are $\mu \in (0,1)$ and $\nu \in (0,\mathrm{\infty })$.

Cumulative distribution function

The cumulative distribution function is

$F(x;\mu ,\nu )=\frac{1}{1+{\left(\frac{\mu }{1-\mu }\frac{1-x}{x}\right)}^{\nu }}$

Probability density function

The probability density function is

$f(x;\mu ,\nu )=\frac{\nu }{x(1-x)}\frac{Z(x;\mu ,\nu )}{(1+Z(x;\mu ,\nu ){)}^2}$

where

$Z(x;\mu ,\nu )={\left(\frac{\mu }{1-\mu }\frac{1-x}{x}\right)}^{\nu }$

Quantile function

The quantile function is

$Q(p;\mu ,\nu )=\frac{1}{1+\frac{1-\mu }{\mu }{\left(\frac{1-p}{p}\right)}^{\frac{1}{\nu }}}$

Properties

Support

This distribution has support on the unit interval $(0,1)$. It can be extended to the interval $(a,b)$ by using the random variable

$y=a+(b-a)x$

Median

The median is $Q(\frac{1}{2};\mu ,\nu )=\mu$. Note that the median does not depend on $\nu$.

Simulation

One can sample from this distribution using inverse transform sampling: Let $p\sim U(0,1)$, where the latter is the standard uniform distribution. Then $x=Q(p;\mu ,\nu )$ follows this distribution.

Symmetry

The distribution is symmetric when $\mu =\frac{1}{2}$. In that case, it is furthermore bell-shaped when $\nu >1$ and bathtub-shaped when $\nu <1$.

Special values

• $\mu =\frac{1}{2},\nu =1$ yields the standard uniform distribution.

• $\nu \to \mathrm{\infty }$ approaches a delta distribution centered at $\mu$.

• $\nu \to 0$ approaches a Bernoulli distribution with parameter $\frac{1}{2}$.

Relation to logistic distribution

Let $\sigma$ be the logistic function and ${\sigma }^{-1}$ be the logit function, its inverse.

Let $y$ follow the logistic distribution with location parameter ${\sigma }^{-1}(\mu )$ and scale parameter ${\nu }^{-1}$. Then $x=\sigma (y)$ follows this distribution.

Applications

On the Birnbaum and Saunders dataset and the Aarset dataset on fatigue and lifetimes of industrial components and materials, the LU distribution is more flexible than various alternative distributions, in the sense of achieving lower values of the Akaike information criterion and Bayesian information criterion.

See also

• Beta distribution
• Kumaraswamy distribution
• Logit-normal distribution
• Logistic distribution
• Metalog distribution
• Distributions supported on a bounded interval

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