Automatic differentiation of Box-Cox transformations with application to random effects models for continuous non-normal response

The Box-Cox transformation generates a family of distributions used in the modelling of continuous, non-Normal data. When data are grouped, the unknown parameters of the distribution are modelled as depending on random effects, and inference by Laplace-approximate marginal likelihood involves challenging computations that are aided by modern advances in automatic differentiation methods. Although mathematically continuously differentiable everywhere, the removable singularity present at zero in the standard form of the Box-Cox transformation renders it incompatible with automatic differentiation methods, complicating implementation of Laplace-approximate marginal likelihood. In this paper we consider two automatically-differentiable approximations to the Box-Cox transformation, with Gauss-Legendre quadrature shown to be especially accurate and fast. Application to Hodges’ famous state-level insurance premiums data is used to illustrate the computational aspects of the approximations. In particular, the flexibility of the implementation allows the use of heavy-tailed random effects distributions with essentially no additional user effort, and this is emphasized when we obtain a 92% reduction in mean-squared error for predicting state-level median premiums, by changing only a single line of code.