A Censored Transformed Model for Proportional Outcomes with Boundary Mass and an Application to Loss Given Default Modeling

2026-06-19

AI Summary

This study addresses ratio-type response variables—such as Loss Given Default (LGD)—that exhibit point masses at the boundaries 0 and 1. We propose the Zero-One Censored Transformed Normal (ZOC-TN) model, which uniquely combines an affine-logit transformation with a truncated normal distribution to flexibly capture density shapes within the open interval (0,1). The framework accommodates nonlinear covariate effects and spatiotemporal residual structures while maintaining modeling flexibility, parsimony, and numerical stability. Notably, ZOC-TN integrates seamlessly into both gradient boosting trees and Gaussian process frameworks. Empirical evaluation on a large-scale U.S. residential mortgage LGD dataset demonstrates that a tree-boosted ZOC-TN model augmented with a spatiotemporal frailty Gaussian process significantly outperforms existing benchmarks, underscoring the critical importance of nonlinear effects and spatiotemporal heterogeneity in LGD modeling.

Abstract

We introduce the zero-one censored transformed normal (ZOC-TN) model for proportional responses with potential probability mass at the boundaries 0 and 1. The model combines a censored Gaussian variable with a two-parameter affine-logit transformation on the interior (0,1). We characterize the transformation parameters, establish large-sample properties, and relate the affine-logit specification to broader classes of interior distributions. Theoretical and experimental results demonstrate that the proposed model can capture a wider range of qualitative density shapes than several benchmark models while remaining parsimonious, computationally efficient, and numerically stable. Furthermore, the ZOC-TN model can be extended (i) to account for nonlinearities and interactions in a tree-boosting machine learning framework and (ii) to explicitly model residual spatio-temporal variability. We apply the ZOC-TN model to loss given default (LGD) modeling for a large dataset of U.S. residential mortgages and compare it to multiple benchmark models. We find that a tree-boosted ZOC-TN model with a spatio-temporal frailty Gaussian process delivers the strongest out-of-sample performance, indicating that mortgage losses are shaped by nonlinear covariate effects and by unaccounted-for space-time variation.

Problem

• proportional outcomes
• boundary mass
• loss given default
• censored data
• spatio-temporal variability

Innovation

• zero-one censored
• affine-logit transformation
• tree-boosting
• spatio-temporal frailty
• proportional outcomes