Maximum likelihood estimation of normal-gamma and normal-Nakagami stochastic frontier models

The gamma and Nakagami distributions have an advantage over other proposed flexible inefficiency distributions in that they can accommodate not only non-zero modes, but also cases in which many firms lie arbitrarily close to the frontier. We propose a normal-Nakagami stochastic frontier model, which provides a generalisation of the normal-half normal that is more flexible than the familiar normal-truncated normal. The normal-gamma model has already attracted much attention, but estimation and efficiency prediction have relied on approximation methods. We derive exact expressions for likelihoods and efficiency predictors, and demonstrate direct maximum likelihood estimation of both models. Across three empirical applications, we show that the models avoid a convergence issue that affects the normal-truncated normal model, and can accommodate a concentration of observations near the frontier similar to zero-inefficiency stochastic frontier models. We provide Python implementations via the FronPy package.