The logistic distribution receives its name from its
cumulative distribution function, which is an instance of the family of logistic functions. The cumulative distribution function of the logistic distribution is also a scaled version of the
hyperbolic tangent.
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:
Alternative parameterization
An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, , in terms of the standard deviation, , using the substitution , where . The alternative forms of the above functions are reasonably straightforward.
One of the most common applications is in
logistic regression, which is used for modeling
categoricaldependent variables (e.g., yes-no choices or a choice of 3 or 4 possibilities), much as standard
linear regression is used for modeling
continuous variables (e.g., income or population). Specifically, logistic regression models can be phrased as
latent variable models with
error variables following a logistic distribution. This phrasing is common in the theory of
discrete choice models, where the logistic distribution plays the same role in logistic regression as the
normal distribution does in
probit regression. Indeed, the logistic and normal distributions have a quite similar shape. However, the logistic distribution has
heavier tails, which often increases the
robustness of analyses based on it compared with using the normal distribution.
Physics
The PDF of this distribution has the same functional form as the derivative of the
Fermi function. In the theory of electron properties in semiconductors and metals, this derivative sets the relative weight of the various electron energies in their contributions to electron transport. Those energy levels whose energies are closest to the distribution's "mean" (
Fermi level) dominate processes such as electronic conduction, with some smearing induced by temperature.[3]: 34 Note however that the pertinent probability distribution in
Fermi–Dirac statistics is actually a simple
Bernoulli distribution, with the probability factor given by the Fermi function.
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
hydrology the distribution of long duration river discharge and rainfall (e.g., monthly and yearly totals, consisting of the sum of 30 respectively 360 daily values) is often thought to be almost normal according to the
central limit theorem.[5] The
normal distribution, however, needs a numeric approximation. As the logistic distribution, which can be solved analytically, is similar to the normal distribution, it can be used instead. The blue picture illustrates an example of fitting the logistic distribution to ranked October rainfalls—that are almost normally distributed—and it shows the 90%
confidence belt based on the
binomial distribution. The rainfall data are represented by
plotting positions as part of the
cumulative frequency analysis.
Chess ratings
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).
The
metalog distribution is generalization of the logistic distribution, in which power series expansions in terms of are substituted for logistic parameters and . 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:
This integral is well-known[6] and can be expressed in terms of
Bernoulli numbers:
^Davies, John H. (1998). The Physics of Low-dimensional Semiconductors: An Introduction. Cambridge University Press.
ISBN9780521484916.
^A. Di Crescenzo, B. Martinucci (2010) "A damped telegraph random process with logistic stationary distribution", J. Appl. Prob., vol. 47, pp. 84–96.
^Ritzema, H.P., ed. (1994).
Frequency and Regression Analysis. Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp.
175–224.
ISBN90-70754-33-9.
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.
ISBN0-8247-8587-8.
Modis, Theodore (1992) Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York.
ISBN0-671-75917-5