Generalized logistic distribution

The term generalized logistic distribution is used as the name for several different families of probability distributions. For example, Johnson et al. list four forms, which are listed below.

Type I has also been called the skew-logistic distribution. Type IV subsumes the other types and is obtained when applying the logit transform to beta random variates. Following the same convention as for the log-normal distribution, type IV may be referred to as the logistic-beta distribution, with reference to the standard logistic function, which is the inverse of the logit transform.

For other families of distributions that have also been called generalized logistic distributions, see the shifted log-logistic distribution, which is a generalization of the log-logistic distribution; and the metalog ("meta-logistic") distribution, which is highly shape-and-bounds flexible and can be fit to data with linear least squares.

Definitions

The following definitions are for standardized versions of the families, which can be expanded to the full form as a location-scale family. Each is defined using either the cumulative distribution function (F) or the probability density function (ƒ), and is defined on (-∞,∞).

Type I

$F(x;\alpha )={\frac {1}{(1+e^{-x})^{\alpha }}}\equiv (1+e^{-x})^{-\alpha },\quad \alpha >0.$

The corresponding probability density function is:

$f(x;\alpha )={\frac {\alpha e^{-x}}{\left(1+e^{-x}\right)^{\alpha +1}}},\quad \alpha >0.$

This type has also been called the "skew-logistic" distribution.

Type II

$F(x;\alpha )=1-{\frac {e^{-\alpha x}}{(1+e^{-x})^{\alpha }}},\quad \alpha >0.$

The corresponding probability density function is:

$f(x;\alpha )={\frac {\alpha e^{-\alpha x}}{(1+e^{-x})^{\alpha +1}}},\quad \alpha >0.$

Type III

$f(x;\alpha )={\frac {1}{B(\alpha ,\alpha )}}{\frac {e^{-\alpha x}}{(1+e^{-x})^{2\alpha }}},\quad \alpha >0.$

Here B is the beta function. The moment generating function for this type is

$M(t)={\frac {\Gamma (\alpha -t)\Gamma (\alpha +t)}{(\Gamma (\alpha ))^{2}}},\quad -\alpha <t<\alpha .$

The corresponding cumulative distribution function is:

$F(x;\alpha )={\frac {\left(e^{x}+1\right)\Gamma (\alpha )e^{\alpha (-x)}\left(e^{-x}+1\right)^{-2\alpha }\,_{2}{\tilde {F}}_{1}\left(1,1-\alpha ;\alpha +1;-e^{x}\right)}{B(\alpha ,\alpha )}},\quad \alpha >0.$

Type IV

${\begin{aligned}f(x;\alpha ,\beta )&={\frac {1}{B(\alpha ,\beta )}}{\frac {e^{-\beta x}}{(1+e^{-x})^{\alpha +\beta }}},\quad \alpha ,\beta >0\\[4pt]&={\frac {\sigma (x)^{\alpha }\sigma (-x)^{\beta }}{B(\alpha ,\beta )}}.\end{aligned}}$

Where, B is the beta function and $\sigma (x)=1/(1+e^{-x})$ is the standard logistic function. The moment generating function for this type is

$M(t)={\frac {\Gamma (\beta -t)\Gamma (\alpha +t)}{\Gamma (\alpha )\Gamma (\beta )}},\quad -\alpha <t<\beta .$

This type is also called the "exponential generalized beta of the second type".

The corresponding cumulative distribution function is:

$F(x;\alpha ,\beta )={\frac {\left(e^{x}+1\right)\Gamma (\alpha )e^{\beta (-x)}\left(e^{-x}+1\right)^{-\alpha -\beta }\,_{2}{\tilde {F}}_{1}\left(1,1-\beta ;\alpha +1;-e^{x}\right)}{B(\alpha ,\beta )}},\quad \alpha ,\beta >0.$

Relationship between types

Type IV is the most general form of the distribution. The Type III distribution can be obtained from Type IV by fixing $\beta =\alpha$. The Type II distribution can be obtained from Type IV by fixing $\alpha =1$ (and renaming $\beta$ to $\alpha$). The Type I distribution can be obtained from Type IV by fixing $\beta =1$. Fixing $\alpha =\beta =1$ gives the standard logistic distribution.

Type IV (logistic-beta) properties

Type IV probability density functions (means=0, variances=1)

The Type IV generalized logistic, or logistic-beta distribution, with support $x\in \mathbb {R}$ and shape parameters $\alpha ,\beta >0$, has (as shown above) the probability density function (pdf):

$f(x;\alpha ,\beta )={\frac {1}{B(\alpha ,\beta )}}{\frac {e^{-\beta x}}{(1+e^{-x})^{\alpha +\beta }}}={\frac {\sigma (x)^{\alpha }\sigma (-x)^{\beta }}{B(\alpha ,\beta )}},$

where $\sigma (x)=1/(1+e^{-x})$ is the standard logistic function. The probability density functions for three different sets of shape parameters are shown in the plot, where the distributions have been scaled and shifted to give zero means and unity variances, in order to facilitate comparison of the shapes.

In what follows, the notation $B_{\sigma }(\alpha ,\beta )$ is used to denote the Type IV distribution.

Relationship with Gamma Distribution

This distribution can be obtained in terms of the gamma distribution as follows. Let $y\sim {\text{Gamma}}(\alpha ,\gamma )$ and independently, $z\sim {\text{Gamma}}(\beta ,\gamma )$ and let $x=\ln y-\ln z$. Then $x\sim B_{\sigma }(\alpha ,\beta )$.

Symmetry

If $x\sim B_{\sigma }(\alpha ,\beta )$, then $-x\sim B_{\sigma }(\beta ,\alpha )$.

Mean, variance and skewness

By using the logarithmic expectations of the gamma distribution, the mean and variance can be derived as:

${\begin{aligned}{\text{E}}[x]&=\psi (\alpha )-\psi (\beta )\\{\text{var}}[x]&=\psi '(\alpha )+\psi '(\beta )\\\end{aligned}}$

where $\psi$ is the digamma function, while $\psi '$ is its first derivative, also known as the trigamma function. Similarly, the skewness can be expressed in terms of the tetragamma function:

${\text{skew}}[x]=\psi ''(\alpha )-\psi ''(\beta )$

The sign (and therefore the handedness) of the skewness is the same as the sign of $\alpha -\beta$.

Mode

The mode (pdf maximum) can be derived by finding $x$ where the log pdf derivative is zero:

${\frac {d}{dx}}\ln f(x;\alpha ,\beta )=\alpha \sigma (-x)-\beta \sigma (x)=0$

This simplifies to $\alpha /\beta =e^{x}$, so that:

${\text{mode}}[x]=\ln {\frac {\alpha }{\beta }}$

Tail behaviour

Tail comparison: Type IV (means=0, variances=1) vs standard normal, vs standard Cauchy

In each of the left and right tails, one of the sigmoids in the pdf saturates to one, so that the tail is formed by the other sigmoid. For large negative $x$, the left tail of the pdf is proportional to $\sigma (x)^{\alpha }\approx e^{\alpha x}$, while the right tail (large positive $x$) is proportional to $\sigma (-x)^{\beta }\approx e^{-\beta x}$. This means the tails are indepently controlled by $\alpha$ and $\beta$. Although type IV tails are heavier than those of the normal distribution ($e^{-{\frac {x^{2}}{2v}}}$, for variance $v$), the type IV means and variances remain finite for all $\alpha ,\beta >0$. This is in contrast with the Cauchy distribution for which the mean and variance do not exist. In the log pdf plots shown here, the type IV tails are linear, the normal distribution tails are quadratic and the Cauchy tails are logarithmic.

Exponential family properties

$B_{\sigma }(\alpha ,\beta )$ forms an exponential family with natural parameters $\alpha$ and $\beta$ and sufficient statistics $\log \sigma (x)$ and $\log \sigma (-x)$. The expected values of the sufficient statistics can be found by differentiation of the log-normalizer:

${\begin{aligned}E[\log \sigma (x)]&={\frac {\partial \log B(\alpha ,\beta )}{\partial \alpha }}=\psi (\alpha )-\psi (\alpha +\beta )\\E[\log \sigma (-x)]&={\frac {\partial \log B(\alpha ,\beta )}{\partial \beta }}=\psi (\beta )-\psi (\alpha +\beta )\\\end{aligned}}$

Given a data set $x_{1},\ldots ,x_{n}$ assumed to have been generated IID from $B_{\sigma }(\alpha ,\beta )$, the maximum-likelihood parameter estimate is:

${\begin{aligned}{\hat {\alpha }},{\hat {\beta }}=\arg \max _{\alpha ,\beta }&\;{\frac {1}{n}}\sum _{i=1}^{n}\log f(x_{i};\alpha ,\beta )\\=\arg \max _{\alpha ,\beta }&\;\alpha {\Bigl (}{\frac {1}{n}}\sum _{i}\log \sigma (x_{i}){\Bigr )}+\beta {\Bigl (}{\frac {1}{n}}\sum _{i}\log \sigma (-x_{i}){\Bigr )}-\log B(\alpha ,\beta )\\=\arg \max _{\alpha ,\beta }&\;\alpha \,{\overline {\log \sigma (x)}}+\beta \,{\overline {\log \sigma (-x)}}-\log B(\alpha ,\beta )\end{aligned}}$

where the overlines denote the averages of the sufficient statistics. The maximum-likelihood estimate depends on the data only via these average statistics. Indeed, at the maximum-likelihood estimate the expected values and averages agree:

${\begin{aligned}\psi ({\hat {\alpha }})-\psi ({\hat {\alpha }}+{\hat {\beta }})&={\overline {\log \sigma (x)}}\\\psi ({\hat {\beta }})-\psi ({\hat {\alpha }}+{\hat {\beta }})&={\overline {\log \sigma (-x)}}\\\end{aligned}}$

which is also where the partial derivatives of the above maximand vanish.

Relationships with other distributions

Relationships with other distributions include:

• The log-ratio of gamma variates is of type IV as detailed above.
• If $y\sim {\text{BetaPrime}}(\alpha ,\beta )$, then $x=\ln y$ has a type IV distribution, with parameters $\alpha$ and $\beta$. See beta prime distribution.
• If $z\sim {\text{Gamma}}(\beta ,1)$ and $y\mid z\sim {\text{Gamma}}(\alpha ,z)$, where $z$ is used as the rate parameter of the second gamma distribution, then $y$ has a compound gamma distribution, which is the same as ${\text{BetaPrime}}(\alpha ,\beta )$, so that $x=\ln y$ has a type IV distribution.
• If $p\sim {\text{Beta}}(\alpha ,\beta )$, then $x={\text{logit}}\,p$ has a type IV distribution, with parameters $\alpha$ and $\beta$. See beta distribution. The logit function, $\mathrm {logit} (p)=\log {\frac {p}{1-p}}$ is the inverse of the logistic function. This relationship explains the name logistic-beta for this distribution: if the logistic function is applied to logistic-beta variates, the transformed distribution is beta.

Large shape parameters

Type IV vs normal distribution with matched mean and variance. For large values of $\alpha ,\beta$, the pdf's are very similar, except for very rare values of $x$.

For large values of the shape parameters, $\alpha ,\beta \gg 1$, the distribution becomes more Gaussian. This is demonstrated in the pdf and log pdf plots here.

Random variate generation

Since random sampling from the gamma and beta distributions are readily available on many software platforms, the above relationships with those distributions can be used to generate variates from the type IV distribution.

Generalization with location and scale parameters

A flexible, four-parameter family can be obtained by adding location and scale parameters. One way to do this is if $x\sim B_{\sigma }(\alpha ,\beta )$, then let $y=kx+\delta$, where $k>0$ is the scale parameter and $\delta \in \mathbb {R}$ is the location parameter. The four-parameter family obtained thus has the desired additional flexibility, but the new parameters may be hard to interpret because $\delta \neq E[y]$ and $k^{2}\neq {\text{var}}[y]$. Moreover maximum-likelihood estimation with this parametrization is hard. These problems can be addressed as follows.

Recall that the mean and variance of $x$ are:

${\begin{aligned}{\tilde {\mu }}&=\psi (\alpha )-\psi (\beta ),&{\tilde {s}}^{2}&=\psi '(\alpha )+\psi '(\beta )\end{aligned}}$

Now expand the family with location parameter $\mu \in \mathbb {R}$ and scale parameter $s>0$, via the transformation:

${\begin{aligned}y&=\mu +{\frac {s}{\tilde {s}}}(x-{\tilde {\mu }})\iff x={\tilde {\mu }}+{\frac {\tilde {s}}{s}}(y-\mu )\end{aligned}}$

so that $\mu =E[y]$ and $s^{2}={\text{var}}[y]$ are now interpretable. It may be noted that allowing $s$ to be either positive or negative does not generalize this family, because of the above-noted symmetry property. We adopt the notation $y\sim {\bar {B}}_{\sigma }(\alpha ,\beta ,\mu ,s^{2})$ for this family.

If the pdf for $x\sim B_{\sigma }(\alpha ,\beta )$ is $f(x;\alpha ,\beta )$, then the pdf for $y\sim {\bar {B}}_{\sigma }(\alpha ,\beta ,\mu ,s^{2})$ is:

${\bar {f}}(y;\alpha ,\beta ,\mu ,s^{2})={\frac {\tilde {s}}{s}}\,f(x;\alpha ,\beta )$

Maximum-likelihood estimation for this family is discussed below.

Maximum likelihood parameter estimation

Since the (logarithms of) the logistic and beta functions are readily available in software packages with automatic differentiation, gradients of the log-pdf with respect to the parameters can be easily obtained, so that gradient-based numerical optimization can be used to make maximum likelihood estimates of the parameters of this distribution.

See also

• Champernowne distribution, another generalization of the logistic distribution.