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)
With the rapid development of machine learning and deep learning techniques, actuaries and the broader insurance industry face a persistent trade-off between predictive accuracy and interpretability. This paper provides a comprehensive applied assessment of Explainable Boosting Machines (EBM) in a car insurance framework, focusing on claim frequency and severity modeling. EBM combines the additive structure of generalized additive models (GAM) with a cyclic gradient boosting algorithm, resulting in a glass-box model whose predictions are interpretable by design. Using real-world data, we empirically illustrate its practical relevance and compare EBM with modern benchmark models used in non-life insurance pricing. The evaluation considers (i) out-of-sample predictive accuracy, including Murphy diagrams and Bregman dominance tests, and (ii) calibration assessment using T-reliability diagrams and Murphy's score decomposition. Finally, we highlight the link between EBM predictions and Shapley values, showing how predictions can be transparently decomposed into exact main and pairwise interaction effects, providing actionable insights beyond predictive performance.
Markéta Krúpová, Nabil Rachdi, Quentin Guibert
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
We develop a statistical learning theory for gradient boosting applied to the estimation of covariate-dependent Generalized Pareto (GP) distributions in the context of Peaks-over-Threshold modeling. After an orthogonal reparametrization of the GP likelihood that diagonalizes its Fisher information matrix, we cast the estimation problem within the Empirical Risk Minimization (ERM) framework and derive non-asymptotic error bounds for the boosting estimator. Our analysis accounts for three distinct sources of error in the process: statistical fluctuations, the approximation bias inherent to the asymptotic nature of the GP model-controlled under second-order regular variation-and the approximation error associated with the finite number of boosting iterates, making explicit the resulting bias-variance trade-off. We illustrate the practical benefits of the reparametrization through simulations, showing that it significantly reduces gradient correlation during training and improves convergence stability. The methodology is applied to a medical malpractice insurance dataset from the Texas Department of Insurance, comprising over 18 000 closed claims. The gradient boosting approach yields a good fit for the tail of settlement cost distributions and reveals that the number of days to settlement is the dominant predictor of tail heaviness, consistent with earlier findings in the reserving literature.
Stéphane Lhaut, Olivier Lopez
1CREST, CNRS, Ecole polytechnique, Groupe ENSAE-ENSAI, ENSAE Paris, Institut Polytechnique de Paris, Palaiseau, France
We introduce CopFITi, a copula model for probabilistic forecasting of irregular multivariate time series (IMTS). Our model combines the expressivity of normalizing flows for univariate marginals with the consistency and flexibility of a Gaussian Mixture Copula for the joint dependency structure. Our experiments show that copula-based approaches, which decouple the marginals from the joint, yield better marginal models than architectures that directly fit the full joint. With CopFITi, we propose the first IMTS copula that is marginalization-consistent by construction and establish a new state of the art in joint IMTS density modeling.
Christian Klötergens, Tom Hanika, Lars Schmidt-Thieme +1
Institute of Computer Science University of Hildesheim Hildesheim, Germany