A general framework for computation and estimation using the saddlepoint approximation
The saddlepoint approximation provides highly accurate approximations to probability density and mass functions using only the corresponding moment generating functions (MGFs). Recent work has increasingly seen the saddlepoint approximation applied to likelihood functions, enabling likelihood-based inference in models where exact likelihoods are intractable. However, existing implementations have largely been developed on a model-by-model basis, and the methodology remains underutilized because of the conceptual and computational challenges of working with MGFs. We introduce a unified framework for model construction and computation using the saddlepoint approximation. The framework is based on a collection of model-building operations that preserve access to MGFs while allowing complex distributions to be constructed from simpler components. With these components, users need only provide a high-level specification of the model structure, from which the software automatically assembles the necessary generating functions, saddlepoints, and gradients, and performs the optimization of the saddlepoint likelihood. We also introduce a diagnostic that quantifies the difference between saddlepoint and exact likelihood estimates, even when the exact likelihood is unavailable. The framework is implemented in the R package saddlepoint and provides fast, convenient computation of parameter estimates, standard errors, and the discrepancy diagnostic. Numerous examples illustrate the scope and flexibility of the approach.