Joint Distributional Copula Regression for Insurance Claim Frequency and Severity via Boosting
Authors: Banghee So, Na Zhang
Organizations: Department of Mathematics, Towson University, 7800 York Rd RM335, Towson, 21204, MD, USA. · Department of Mathematics, Towson University, 7800 York Rd RM371, Towson, 21204, MD, USA.
In this paper, we propose a model-based boosting approach for fitting structured additive distributional copula regression models with bivariate responses, focusing particularly on the joint modeling of claim frequency and claim severity. The proposed approach is based on generalized additive models for location, scale, and shape (GAMLSS), which extend the general idea of GAMs by allowing multiple parameters of a response distribution, rather than only its conditional mean, to depend on covariates. The model is therefore intrinsically interpretable at the level of its additive predictors, providing greater transparency and auditability than black-box alternatives. We evaluate the proposed framework using synthetic and real-world data. We demonstrate its ability to recover the underlying covariate effects and dependence structure, identify informative variables, and achieve competitive predictive performance relative to benchmark models. We also illustrate how the fitted model can be interpreted.
Table 1: Predictive performance of the proposed model and two benchmark models across different dimensional settings
Model
Negative log-likelihood
Energy score
Frank copula boosting
1,293 (24)
1.40 (0.04)
Gaussian copula boosting
1,314 (26)
1.41 (0.04)
Gumbel copula boosting
1,323 (32)
1.41 (0.05)
Table 2: Effect of copula misspecification on predictive performance for p=10
Figure 2: Partial effects for the seven informative covariates. Blue dashed lines denote the true effects, gray lines the estimates from the 10 replicated datasets, and black solid lines their average.
Figure 3: Conditional Gamma cumulative distribution functions and quantiles for varying values of the selected covariates, using parameters estimated via the proposed copula-based distributional model.
Figure 4: Top 10 covariates based on the mean variable selection rate across 10 replications. Panels (a)–(d) correspond to predictors for η1(1) , η1(2) , η2(2) , and η(c) , respectively.
Model
Negative log-likelihood
Energy score
Frank copula boosting
-305,276
130.8
Gaussian copula boosting
-305,228
131.0
Gumbel copula boosting
-305,065
131.3
Univariate boosting
-302,762
134.4
Univariate MLE
-297,023
132.6
Table 3: Predictive performance on the synthetic telematics dataset.
Figure 5: Covariates included in the predictor for each parameter, ranked by selection rate.
Figure 6: Effects of covariates on Kendall’s tau rank correlation coefficient between claim frequency and severity
Figure 7: Estimated partial effects on the additive predictor for the Poisson mean
Figure 8: Conditional Gamma cumulative distributions and quantiles of claim severity for varying values of the informative covariates.
Figure 9: Joint probabilities from the copula model separated by low risk (grey) and high risk (white)
Mar 27, 2025·Markéta Krúpová, Nabil Rachdi, Quentin GuibertInterpretable ML
CEREMADE · 1ADDACTIS France, 46 bis Chemin du Vieux Moulin, 69160 Tassin-la-Demi-Lune, France · 2CEREMADE, Universit´e Paris-Dauphine, PSL University, CNRS, 75016 Paris, France