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 t is the standard logistic function, then X = t(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 (PDF) of a logit-normal distribution, for 0 < x < 1, is:
f X ( x ; μ , σ ) = 1 σ 2 π 1 x ( 1 − x ) e − ( logit ( x ) − μ ) 2 2 σ 2 {\displaystyle f_{X}(x;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\,{\frac {1}{x(1-x)}}\,e^{-{\frac {(\operatorname {logit} (x)-\mu )^{2}}{2\sigma ^{2}}}}}
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. The moments can be estimated by numerical integration, however numerical integration can be prohibitive when the values of μ , σ 2 {\textstyle \mu ,\sigma ^{2}} are such that the density function diverges to infinity at the end points zero and one. An alternative is to use the observation that the logit-normal is a transformation of a normal random variable. This allows us to approximate the n {\displaystyle n} -th moment via the following quasi Monte Carlo estimate E [ X n ] ≈ 1 K − 1 ∑ i = 1 K − 1 ( P ( Φ μ , σ 2 − 1 ( i / K ) ) ) n , {\displaystyle E[X^{n}]\approx {\frac {1}{K-1}}\sum _{i=1}^{K-1}\left(P\left(\Phi _{\mu ,\sigma ^{2}}^{-1}(i/K)\right)\right)^{n},}
where P {\textstyle P} is the standard logistic function, and Φ μ , σ 2 − 1 {\textstyle \Phi _{\mu ,\sigma ^{2}}^{-1}} is the inverse cumulative distribution function of a normal distribution with mean and variance μ , σ 2 {\textstyle \mu ,\sigma ^{2}} . When n = 1 {\displaystyle n=1} , this corresponds to the mean.
Mode or modes When the derivative of the density equals 0 then the location of the mode x satisfies the following equation:
logit ( x ) = σ 2 ( 2 x − 1 ) + μ . {\displaystyle \operatorname {logit} (x)=\sigma ^{2}(2x-1)+\mu .}
For some values of the parameters there are two solutions, i.e. the distribution is bimodal.
Multivariate generalization The logistic normal distribution is a generalization of the logit–normal distribution to D-dimensional probability vectors by taking a logistic transformation of a multivariate normal distribution.
Probability density function The probability density function is:
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