arXiv subjects

Mario V. Wüthrich

Publications and source records attributed to Mario V. Wüthrich.

At least 19 recordsLinked to original sources

arXiv2026

A Practical Guide on Graphical Model Validation

This manuscript formalizes the most popular model validation tools used in general insurance actuarial modeling. These include graphical tools like calibration plots, actual-vs-expected plots, lift charts, Murphy diagrams, as well as classical statistical tools such as Bregman losses, deviance losses, elementary losses, Murphy's decomposition and Gini scores. Particular emphasis is placed on whether calibration and discrimination are studied under a policy-weighted or an exposure-weighted population measure. This distinction is crucial in ensuring that premium schemes are calibrated on the correct scale.

stat.ML↗

arXiv2026

Many Optimizers But Only One Training Path: Repeated Resampling for Adaptive Optimizer Selection

An optimizer is usually chosen before training a deep neural network and then kept fixed. Treating optimizer choice as a hyperparameter could boost performance, but it requires several complete training runs and discards all but the winner. Repeated Optimizer Resampling (ROR) instead searches during one evolving run. Every $b$ epochs, each candidate optimizer scouts from the current model weights for $s$ epochs. The best scout continues for the remaining $b-s$ epochs, and that completed segment becomes the new incumbent if it improves the validation objective. This design allows the preferred optimizer to change as training progresses. We compare two variants of ROR on MNIST, Fashion-MNIST, and two motor insurance claim-count models. Nine fixed optimizers and both ROR variants are evaluated with the same ten seeds. One-epoch ROR uses 24\% to 35\% of the aggregate training needed to identify the best fixed optimizer exhaustively and remains close to that optimizer on all four tasks. These results support short scouting as a practical way to search over optimizers without completing every candidate run.

cs.LG↗

arXiv2026

The Balance Property: The Constrained Case, with a View on Risk Sharing

The balance property is an important property of fitted statistical models deployed for insurance pricing. It guarantees that the total actuarial price in the fitted model is equal to the totally observed loss used to fit the model. This can be seen as an in-sample global unbiasedness property. Maximum likelihood fitted generalized linear models (GLMs) with canonical links automatically fulfill the balance property. Lindholm-Wüthrich (Scandinavian Actuarial Journal, 2026) discussed two popular balance correction methods in case the balance property fails to hold. This note extends this discussion with a third method, constrained GLM fitting, that turns out to be superior over the two previously discussed ones. Moreover, we highlight the connection between the balance property and ex-post risk sharing rules.

math.ST↗

arXiv2026

Insurance Pricing Optimization via Off-Policy Evaluation

Traditional insurance pricing relies on risk-based principles that ensure actuarial fairness and solvency but do not explicitly account for policyholders' price sensitivity. We formulate insurance pricing as a decision-making problem and study it using tools from off-policy evaluation and stochastic control. We propose a kernelized inverse propensity score estimator that exploits local structure in the action space and yields variance reduction compared to the classical inverse propensity score estimator. Building on these value estimates, we investigate policy optimization and present two practical approaches for computing optimal pricing rules: an interpretable data-shared Lasso formulation and a flexible policy parameterization based on neural networks. Using a controlled synthetic travel insurance environment, we empirically confirm the theoretical results and show that neural networks outperform existing techniques for policy optimization.

stat.ML↗

arXiv2026

State-Space Representation of INGARCH Models and Their Application in Insurance

Integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) models are a popular framework for modeling serial dependence in count time-series. While convenient for modeling, prediction, and estimation, INGARCH models lack a clear theoretical justification for the evolution step. This limitation not only makes interpretation difficult and complicates the inclusion of covariates, but can also make the handling of missing data computationally burdensome. Consequently, applying such models in an insurance context, where covariates and missing observations are common, can be challenging. In this paper, we first introduce the marginalized state-space model (M-SSM), defined solely through the marginal distribution of the observations, and show that INGARCH models arise as special cases of this framework. The M-SSM formulation facilitates the natural incorporation of covariates and missing data mechanisms, and this representation in turn provides a coherent way to incorporate these elements within the INGARCH model as well. We then demonstrate that an M-SSM can admit an observation-driven state-space model (O-SSM) representation when suitable assumptions are imposed on the evolution of its conditional mean. This lifting from an M-SSM to an O-SSM provides a natural setting for establishing weak stationarity, even in the presence of heterogeneity and missing observations. The proposed ideas are illustrated through the Poisson and the Negative-Binomial INGARCH(1,1) models, highlighting their applicability in predictive analysis for insurance data.

stat.ME↗

arXiv2026

A Note on the Generalized Cape Cod Reserving Method

Claims reserving is one of the most important actuarial tasks in non-life insurance modeling. There are several popular methods to perform claims reserving such as the chain-ladder (CL), the Bornhuetter--Ferguson (BF) or the generalized Cape Cod (GCC) methods. These methods have originally been introduced as deterministic algorithms, and only in a later step, they have been lifted to stochastic models allowing for analyzing claims prediction uncertainty. This holds true for the CL and the BF methods, but not for the GCC method. The purpose of this article is to close this gap and derive an analytical formula for the mean squared error of prediction (MSEP) of the GCC method.

stat.AP↗

arXiv2026

One-Shot Individual Claims Reserving

Individual claims reserving has not yet become established in actuarial practice. We attribute this to the absence of a satisfactory methodology: existing approaches tend to be either overly complex or insufficiently flexible and robust for practical use. Building on the classical chain-ladder (CL) method, we introduced a new perspective on individual claims reserving in Richman and Wüthrich [arXiv:2602.15385]. This manuscript has sparked considerable discussion within the actuarial community. The aim of the present paper is to continue and deepen that discussion, with the ultimate goal of advancing toward a new standard for micro-level reserving.

stat.AP↗

arXiv2026

From Chain-Ladder to Individual Claims Reserving

The chain-ladder (CL) method is the most widely used claims reserving technique in non-life insurance. This manuscript introduces a novel approach to computing the CL reserves based on a fundamental restructuring of the data utilization for the CL prediction procedure. Instead of rolling forward the cumulative claims with estimated CL factors, we estimate multi-period factors that project the latest observations directly to the ultimate claims. This alternative perspective on CL reserving creates a natural pathway for the application of machine learning techniques to individual claims reserving. As a proof of concept, we present a small-scale real data application employing neural networks for individual claims reserving.

stat.AP↗

arXiv2026

In-Context Learning Enhanced Credibility Transformer

The starting point of our network architecture is the Credibility Transformer which extends the classical Transformer architecture by a credibility mechanism to improve model learning and predictive performance. This Credibility Transformer learns credibilitized CLS tokens that serve as learned representations of the original input features. In this paper we present a new paradigm that augments this architecture by an in-context learning mechanism, i.e., we increase the information set by a context batch consisting of similar instances. This allows the model to enhance the CLS token representations of the instances by additional in-context information and fine-tuning. We empirically verify that this in-context learning enhances predictive accuracy by adapting to similar risk patterns. Moreover, this in-context learning also allows the model to generalize to new instances which, e.g., have feature levels in the categorical covariates that have not been present when the model was trained -- for a relevant example, think of a new vehicle model which has just been developed by a car manufacturer.

cs.LG↗

arXiv2026

Calibration Bands for Mean Estimates within the Exponential Dispersion Family

A statistical model is said to be calibrated if the resulting mean estimates perfectly match the true means of the underlying responses. Aiming for calibration is often not achievable in practice as one has to deal with finite samples of noisy observations. A weaker notion of calibration is auto-calibration. An auto-calibrated model satisfies that the expected value of the responses for a given mean estimate matches this estimate. Testing for autocalibration has only been considered recently in the literature and we propose a new approach based on calibration bands. Calibration bands denote a set of lower and upper bounds such that the probability that the true means lie simultaneously inside those bounds exceeds some given confidence level. Such bands were constructed by Yang-Barber (2019) for sub-Gaussian distributions. Dimitriadis et al. (2023) then introduced narrower bands for the Bernoulli distribution. We use the same idea in order to extend the construction to the entire exponential dispersion family that contains for example the binomial, Poisson, negative binomial, gamma and normal distributions. Moreover, we show that the obtained calibration bands allow us to construct various tests for calibration and auto-calibration, respectively. As the construction of the bands does not rely on asymptotic results, we emphasize that our tests can be used for any sample size.

math.ST↗

arXiv2026

Tab-TRM: Tiny Recursive Model for Insurance Pricing on Tabular Data

We introduce Tab-TRM (Tabular-Tiny Recursive Model), a network architecture that adapts the recursive latent reasoning paradigm of Tiny Recursive Models (TRMs) to insurance modeling. Drawing inspiration from both the Hierarchical Reasoning Model (HRM) and its simplified successor TRM, the Tab-TRM model makes predictions by reasoning over the input features. It maintains two learnable latent tokens - an answer token and a reasoning state - that are iteratively refined by a compact, parameter-efficient recursive network. The recursive processing layer repeatedly updates the reasoning state given the full token sequence and then refines the answer token, in close analogy with iterative insurance pricing schemes. Conceptually, Tab-TRM bridges classical actuarial workflows - iterative generalized linear model fitting and minimum-bias calibration - on the one hand, and modern machine learning, in terms of Gradient Boosting Machines, on the other.

cs.LG↗

arXiv2025

Model Monitoring: A General Framework with an Application to Non-life Insurance Pricing

Maintaining the predictive performance of pricing models is challenging when insurance portfolios and data-generating mechanisms evolve over time. Focusing on non-life insurance, we adopt the concept-drift terminology from machine learning and distinguish virtual drift from real concept drift in an actuarial setting. Methodologically, we (i) formalize deviance loss and Murphy's score decomposition to assess global and local auto-calibration; (ii) study the Gini score as a rank-based performance measure, derive its asymptotic distribution, and develop a consistent bootstrap estimator of its asymptotic variance; and (iii) combine these results into a statistically grounded, model-agnostic monitoring framework that integrates a Gini-based ranking drift test with global and local auto-calibration tests. An application to a modified motor insurance portfolio with controlled concept-drift scenarios illustrates how the framework guides decisions on refitting or recalibrating pricing models.

stat.ML↗

arXiv2025

Gini Score under Ties and Case Weights

The Gini score is a popular tool in statistical modeling and machine learning for model validation and model selection. It is a purely rank based score that allows one to assess risk rankings. The Gini score for statistical modeling has mainly been used in a binary context, in which it has many equivalent reformulations such as the receiver operating characteristic (ROC) or the area under the curve (AUC). In the actuarial literature, this rank based score for binary responses has been extended to general real-valued random variables using Lorenz curves and concentration curves. While these initial concepts assume that the risk ranking is generated by a continuous distribution function, we discuss in this paper how the Gini score can be used in the case of ties in the risk ranking. Moreover, we adapt the Gini score to the common actuarial situation of having case weights.

stat.ML↗

arXiv2025

Tree-like Pairwise Interaction Networks

Modeling feature interactions in tabular data remains a key challenge in predictive modeling, for example, as used for insurance pricing. This paper proposes the Tree-like Pairwise Interaction Network (PIN), a novel neural network architecture that explicitly captures pairwise feature interactions through a shared feed-forward neural network architecture that mimics the structure of decision trees. PIN enables intrinsic interpretability by design, allowing for direct inspection of interaction effects. Moreover, it allows for efficient SHapley's Additive exPlanation (SHAP) computations because it only involves pairwise interactions. We highlight connections between PIN and established models such as GA2Ms, gradient boosting machines, and graph neural networks. Empirical results on the popular French motor insurance dataset show that PIN outperforms both traditional and modern neural networks benchmarks in predictive accuracy, while also providing insight into how features interact with each another and how they contribute to the predictions.

stat.ML↗

arXiv2025

Shapley Values: Paired-Sampling Approximations

Originally introduced in cooperative game theory, Shapley values have become a very popular tool to explain machine learning predictions. Based on Shapley's fairness axioms, every input (feature component) gets a credit how it contributes to an output (prediction). These credits are then used to explain the prediction. The only limitation in computing the Shapley values (credits) for many different predictions is of computational nature. There are two popular sampling approximations, sampling KernelSHAP and sampling PermutationSHAP. Our first novel contributions are asymptotic normality results for these sampling approximations. Next, we show that the paired-sampling approaches provide exact results in case of interactions being of maximal order two. Furthermore, the paired-sampling PermutationSHAP possesses the additive recovery property, whereas its kernel counterpart does not.

stat.ML↗

arXiv2024

An Observation-Driven State-Space Model for Claims Size Modeling

State-space models are popular models in econometrics. Recently, these models have gained some popularity in the actuarial literature. The best known state-space models are of Kalman-filter type. These models are so-called parameter-driven because the observations do not impact the state-space dynamics. A second less well-known class of state-space models are so-called observation-driven state-space models where the state-space dynamics is also impacted by the actual observations. A typical example is the Poisson-Gamma observation-driven state-space model for counts data. This Poisson-Gamma model is fully analytically tractable. The goal of this paper is to develop a Gamma- Gamma observation-driven state-space model for claim size modeling. We provide fully tractable versions of Gamma-Gamma observation-driven state-space models, and these versions extend the work of Smith and Miller (1986) by allowing for a fully flexible variance behavior. Additionally, we demonstrate that the proposed model aligns with evolutionary credibility, a methodology in insurance that dynamically adjusts premium rates over time using evolving data.

stat.ME↗

arXiv2024

The Credibility Transformer

Inspired by the large success of Transformers in Large Language Models, these architectures are increasingly applied to tabular data. This is achieved by embedding tabular data into low-dimensional Euclidean spaces resulting in similar structures as time-series data. We introduce a novel credibility mechanism to this Transformer architecture. This credibility mechanism is based on a special token that should be seen as an encoder that consists of a credibility weighted average of prior information and observation based information. We demonstrate that this novel credibility mechanism is very beneficial to stabilize training, and our Credibility Transformer leads to predictive models that are superior to state-of-the-art deep learning models.

cs.LG↗

arXiv2024

Claims processing and costs under capacity constraints

Random delays between the occurrence of accident events and the corresponding reporting times of insurance claims is a standard feature of insurance data. The time lag between the reporting and the processing of a claim depends on whether the claim can be processed without delay as it arrives or whether it remains unprocessed for some time because of temporarily insufficient processing capacity that is shared between all incoming claims. We aim to explain and analyze the nature of processing delays and build-up of backlogs. We show how to select processing capacity optimally in order to minimize claims costs, taking delay-adjusted costs and fixed costs for claims settlement capacity into account. Theoretical results are combined with a large-scale numerical study that demonstrates practical usefulness of our proposal.

q-fin.RM↗