Moments of the logit-normal distribution

Despite the extensive use of the logistic transformation in statistics, the logit transformation of a normal random variable has not been investigated in depth. In particular, it is generally held that moments of a logit-normal random variable must be obtained through numerical integration. We will show, that in general positive integer moments can be constructed using recurrence relations and infinite sums of hyperbolic, exponential and trigonometric functions. We will determine criterion for truncating these infinite sums, while maintaining accuracy and gaining computational efficiency relative to current numerical integration methods for estimating logit-normal moments. We will show all negative moments are exact analytic functions of the moments of the log-normal distribution. Further, given logit-normal moments are known for a subset of possible μ, we will show a relationship exists between log-normal and logit-normal moments for all other possible values of μ, which leads to an exact expression for the first moment when $\mu /\sigma 2$ is an integer.