Optimal Estimator for Logistic Model with Distribution-free Random Intercept

Logistic models with a random intercept are prevalent in medical and social research where clustered and longitudinal data are often collected. Traditionally, the random intercept in these models is assumed to follow some parametric distribution such as the normal distribution. However, such an assumption inevitably raises concerns about model misspecification and misleading inference conclusions, especially when there is dependence between the random intercept and model covariates. To protect against such issues, we use a semiparametric approach to develop a computationally simple and consistent estimator where the random intercept is distribution-free. The estimator is revealed to be optimal and achieve the efficiency bound without the need to postulate or estimate any latent variable distributions. We further characterize other general mixed models where such an optimal estimator exists.