Vector algebra relations

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, which often refers to a multinomial logit version (e.g. ).

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:

$f_{X}(x;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(\operatorname {logit} (x)-\mu )^{2}}{2\sigma ^{2}}}}{\frac {1}{x(1-x)}},\quad x\in (0,1)$

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:

$\operatorname {logit} (x)=\sigma ^{2}(2x-1)+\mu .$

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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