The logistic-normal integral and the moments of the logistic-normal distribution

The logistic-normal integral appears in problems of statistical estimation for logistic models with Gaussian random effects, and generalized linear mixed models.

We study the numerical evaluation of this integral and of its derivatives, and give closed form evaluations at certain points and series expansions.

There is a continuum of possible series expansions, and we single out one series expansion which is optimal for numerical evaluation.

We propose an algorithm for a precise numerical evaluation, based on the optimal series, with good approximation error control in the tails.

As an application we give explicit results for the first two moments of a logistic-normal random variable.