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 mu, we will show a relationship exists between log-normal and logit-normal moments for all other possible values of mu, which leads to an exact expression for the first moment when is an integer.

Automated summary

This research investigates the logit transformation of a normal random variable and proposes methods for constructing positive integer moments using recurrence relations and infinite sums of hyperbolic, exponential, and trigonometric functions. The study also determines criteria for truncating these infinite sums, improving computational efficiency for estimating logit-normal moments. Moreover, it reveals the exact analytic functions of negative moments and establishes a relationship between log-normal and logit-normal moments for different values of the parameter mu, providing an exact expression for the first moment when mu is an integer.