Hidden Truncation Models: Theory and Applications

This review article focuses on hidden truncation models, a versatile framework for modeling a wide range of random phenomena. These models are characterized by the condition that the primary study variable(s) are observable only when a concomitant variable (or a set of concomitant variables in the multivariate case) meets specific criteria. Hidden truncation, where the truncation mechanism is not directly observable and must be inferred from data, adds a layer of complexity that has inspired extensive research. While hidden truncation for normal distributions is straightforward and mathematically manageable, extending hidden truncation models to non-normal distributions often results in complex forms involving computationally complex normalizing constants. This article surveys the foundational development of hidden truncation models, their mathematical and structural properties, and their relationship with the skewing paradigm. Through illustrative examples, we highlight the applicability of hidden truncation models across diverse real-world scenarios. We also examine key aspects of statistical inference, unresolved challenges, and promising directions for future research, offering a comprehensive resource for researchers and practitioners interested in this evolving area of study.