Vector algebra relations
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In probability theory, a logit-normal distribution is a probability distribution of a random variable whose logit has a normal distribution. If Y is a random variable with a normal distribution, and P is the logistic function, then X = P(Y) has a logit-normal distribution; likewise, if X is logit-normally distributed, then Y = logit(X)= log (X/(1-X)) is normally distributed. It is also known as the logistic normal distribution,[1] which often refers to a multinomial logit version (e.g. [2] [3] [4] [5]).
A variable might be modeled as logit-normal if it is a proportion, which is bounded by zero and one, and where values of zero and one never occur.
Characterization
Probability density function
The probability density function of a logit-normal distribution is:
where μ and σ are the mean and standard deviation of the variable’s logit (by definition, the variable’s logit is normally distributed).
The density obtained by changing the sign of μ is symmetrical, in that it is equal to f(1-x;-μ,σ), shifting the mode to the other side of 0.5 (the midpoint of the (0,1) interval).
Moments
The moments of the logit-normal distribution have no analytic solution. However, they can be estimated by numerical integration.
Mode
When the derivative of the density equals 0 then the location of the mode x satisfies the following equation:
See also
- Beta distribution and Kumaraswamy distribution, other two-parameter distributions on a bounded interval with similar shapes
Further reading
- Frederic, P. & Lad, F. (2008) Two Moments of the Logitnormal Distribution. Communications in Statistics-Simulation and Computation. 37: 1263-1269
- Template:Cite JSTOR
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External links
- logitnorm package for R
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- ↑ J Atchison and SM Shen. "Logistic-normal distributions: Some properties and uses." Biometrika, 1980. Google Scholar link
- ↑ http://people.csail.mit.edu/tomasz/papers/huang_hln_tech_report_2006.pdf
- ↑ Peter Hoff, 2003. Link
- ↑ http://www.springerreference.com/docs/html/chapterdbid/205424.html
- ↑ http://brenocon.com/blog/2011/05/log-normal-and-logistic-normal-terminology/