Saddle-point approximations are powerful tools for deriving accurate expressions for densities and distribution functions. This paper provides a foundational introduction to saddle-point approximation and elucidates the underlying principles. Additionally, it reviews the evolution of these techniques from Daniels’ seminal proposal in 1954 to the present day. Our research highlights the promising future of saddle-point approximations, driven by advancements in computational tools, big data, machine learning applications, extensions to more complex models, and increased accessibility through software development. These advancements will significantly broaden the method’s applicability and utility across various scientific and engineering disciplines, paving the way for innovations and breakthroughs in both theoretical and applied research.

saddle-point approximation, density and mass functions, cumulative distribution function

Received: September 4, 2024; Revised: November 5, 2024; Accepted: December 9, 2024; Published: December 30, 2024

How to cite this article: Amal D. Alhejaili and Ateq Ahmed AlGhamedi, A review of saddle-point approximation: theory and applications, Advances and Applications in Statistics 92(3) (2025), 343-390. https://doi.org/10.17654/0972361725015

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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