Logistic distribution

Logistic distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu ,}$ location (real) ${\displaystyle s>0,}$ scale (real)
Support	${\displaystyle x\in (-\infty ,\infty )}$
PDF	${\displaystyle {\frac {e^{-(x-\mu )/s}}{s\left(1+e^{-(x-\mu )/s}\right)^{2}}}}$
CDF	${\displaystyle {\frac {1}{1+e^{-(x-\mu )/s}}}={\frac {1+\tanh {\frac {x-\mu }{2s}}}{2}}}$
Quantile	${\displaystyle \mu +s\log \left({\frac {p}{1-p}}\right)}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle {\frac {s^{2}\pi ^{2}}{3}}}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle 6/5}$
Entropy	${\displaystyle \ln s+2}$
MGF	${\displaystyle e^{\mu t}\mathrm {B} (1-st,1+st)}$ for ${\displaystyle t\in (-1/s,1/s)}$ and ${\displaystyle \mathrm {B} }$ is the Beta function
CF	${\displaystyle e^{it\mu }{\frac {\pi st}{\sinh(\pi st)}}}$
Expected shortfall	${\displaystyle \mu +{\frac {sH(p)}{1-p}}}$ where ${\displaystyle H(p)}$ is the binary entropy function[1] ${\displaystyle H(p)=-p\ln(p)-(1-p)\ln(1-p)}$

In

When the location parameter μ is 0 and the scale parameter s is 1, then the probability density function of the logistic distribution is given by

${\displaystyle {\begin{aligned}f(x;0,1)&={\frac {e^{-x}}{(1+e^{-x})^{2}}}\\[4pt]&={\frac {1}{(e^{x/2}+e^{-x/2})^{2}}}\\[5pt]&={\frac {1}{4}}\operatorname {sech} ^{2}\left({\frac {x}{2}}\right).\end{aligned}}}$

Thus in general the density is:

${\displaystyle {\begin{aligned}f(x;\mu ,s)&={\frac {e^{-(x-\mu )/s}}{s\left(1+e^{-(x-\mu )/s}\right)^{2}}}\\[4pt]&={\frac {1}{s\left(e^{(x-\mu )/(2s)}+e^{-(x-\mu )/(2s)}\right)^{2}}}\\[4pt]&={\frac {1}{4s}}\operatorname {sech} ^{2}\left({\frac {x-\mu }{2s}}\right).\end{aligned}}}$

Because this function can be expressed in terms of the square of the

In this equation μ is the mean, and s is a scale parameter proportional to the standard deviation.

Quantile function

The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function. Its derivative is called the quantile density function. They are defined as follows:

${\displaystyle Q(p;\mu ,s)=\mu +s\ln \left({\frac {p}{1-p}}\right).}$

${\displaystyle Q'(p;s)={\frac {s}{p(1-p)}}.}$

Alternative parameterization

An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, ${\displaystyle s}$, in terms of the standard deviation, ${\displaystyle \sigma }$, using the substitution ${\displaystyle s\,=\,q\,\sigma }$, where ${\displaystyle q\,=\,{\sqrt {3}}/{\pi }\,=\,0.551328895\ldots }$. The alternative forms of the above functions are reasonably straightforward.

Applications

The logistic distribution—and the S-shaped pattern of its

Logistic regression

One of the most common applications is in

The logistic distribution arises as limit distribution of a finite-velocity damped random motion described by a telegraph process in which the random times between consecutive velocity changes have independent exponential distributions with linearly increasing parameters.[4]

Hydrology

In

The United States Chess Federation and FIDE have switched its formula for calculating chess ratings from the normal distribution to the logistic distribution; see the article on Elo rating system (itself based on the normal distribution).

Related distributions

• Logistic distribution mimics the
  sech distribution
  .
• If ${\displaystyle X\sim \mathrm {Logistic} (\mu ,s)}$ then ${\displaystyle kX+\ell \sim \mathrm {Logistic} (k\mu +\ell ,|k|s)}$.
• If ${\displaystyle X\sim }$
  U(0, 1)
  then ${\displaystyle \mu +s(\log X-\log(1-X))\sim \mathrm {Logistic} (\mu ,s)}$.
• If ${\displaystyle X\sim \mathrm {Gumbel} (\mu _{X},\beta )}$ and ${\displaystyle Y\sim \mathrm {Gumbel} (\mu _{Y},\beta )}$ independently then ${\displaystyle X-Y\sim \mathrm {Logistic} (\mu _{X}-\mu _{Y},\beta )\,}$.
• If ${\displaystyle X}$ and ${\displaystyle Y\sim \mathrm {Gumbel} (\mu ,\beta )}$ then ${\displaystyle X+Y\nsim \mathrm {Logistic} (2\mu ,\beta )\,}$ (The sum is not a logistic distribution). Note that ${\displaystyle E(X+Y)=2\mu +2\beta \gamma \neq 2\mu =E\left(\mathrm {Logistic} (2\mu ,\beta )\right)}$.
• If X ~ Logistic(μ, s) then exp(X) ~ LogLogistic${\displaystyle \left(\alpha =e^{\mu },\beta ={\frac {1}{s}}\right)}$, and exp(X) + γ ~ shifted log-logistic${\displaystyle \left(\alpha =e^{\mu },\beta ={\frac {1}{s}},\gamma \right)}$.
• If X ~ Exponential(1) then

${\displaystyle \mu +s\log(e^{X}-1)\sim \operatorname {Logistic} (\mu ,s).}$

• If X, Y ~ Exponential(1) then

${\displaystyle \mu -s\log \left({\frac {X}{Y}}\right)\sim \operatorname {Logistic} (\mu ,s).}$

• The metalog distribution is generalization of the logistic distribution, in which power series expansions in terms of ${\displaystyle p}$ are substituted for logistic parameters ${\displaystyle \mu }$ and ${\displaystyle \sigma }$. The resulting metalog quantile function is highly shape flexible, has a simple closed form, and can be fit to data with linear least squares.

Derivations

Higher-order moments

The nth-order central moment can be expressed in terms of the quantile function:

${\displaystyle {\begin{aligned}\operatorname {E} [(X-\mu )^{n}]&=\int _{-\infty }^{\infty }(x-\mu )^{n}\,dF(x)\\&=\int _{0}^{1}{\big (}Q(p)-\mu {\big )}^{n}\,dp=s^{n}\int _{0}^{1}\left[\ln \!\left({\frac {p}{1-p}}\right)\right]^{n}\,dp.\end{aligned}}}$

This integral is well-known^[6] and can be expressed in terms of Bernoulli numbers:

${\displaystyle \operatorname {E} [(X-\mu )^{n}]=s^{n}\pi ^{n}(2^{n}-2)\cdot |B_{n}|.}$

See also

• generalized logistic distribution
• Tukey lambda distribution
• log-logistic distribution
• half-logistic distribution
• logistic regression
• sigmoid function

Notes

References

Wikimedia Commons has media related to Logistic distribution.

• John S. deCani & Robert A. Stine (1986). "A note on deriving the information matrix for a logistic distribution". The American Statistician. 40. American Statistical Association: 220–222.
  doi:10.2307/2684541
  .
• N., Balakrishnan (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York.
  ISBN 0-8247-8587-8
  .
• Johnson, N. L.; Kotz, S.; N., Balakrishnan (1995). Continuous Univariate Distributions. Vol. 2 (2nd ed.).
  ISBN 0-471-58494-0
  .

• Modis, Theodore (1992) Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York.
  ISBN 0-671-75917-5