Delong, Ł., Wüthrich, M.V., 2025, Universal inference for testing calibration of mean estimates within the Exponential Dispersion Family
Working Paper,
05-12-2025
Calibration of mean estimates for prediction is a crucial property in many applications, particularly in the fields of financial and actuarial decision making. In this paper, we first review classical approaches for validating mean-calibration, and we discuss the Likelihood Ratio Test (LRT) within the Exponential Dispersion Family (EDF). Then, we investigate the framework of universal inference to test for mean-calibration. We develop a sub-sampled split LRT within the EDF that provides finite sample guarantees with universally valid critical values. We investigate type I error, power and e-power of this sub-sampled split LRT, and we compare it to the classical LRT. We propose a novel test statistics based on a sub-sampled split Lq-Likelihood Ratio Test (LqRT) to enhance the performance of the calibration test. A numerical analysis verifies that universal inference with the sub-sampled split LRT and the sub-sampled split LqRT is an attractive alternative to the classical LRT achieving a high power in detecting miscalibration in medium and large samples.
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Delong, Ł., Wüthrich, 2025, Isotonic regression for variance estimation and its role in mean estimation and model validation
Published Paper,
01-07-2024
We study isotonic regression which is a non-parametric rank-preserving regression technique. Under the assumption that the variance function of a response is monotone in its mean functional, we investigate a novel application of isotonic regression as an estimator of this variance function. Our proposal of variance estimation with isotonic regression is used in multiple classical regression problems focused on mean estimation and model validation. In a series of numerical examples, we (1) explore the power variance parameter of the
variance function within Tweedie's family of distributions, (2) derive a semi-parametric bootstrap under heteroskedasticity,
(3) provide a test for auto-calibration, (4) explore a quasi-likelihood approach to benefit from best-asymptotic estimation,
(5) deal with several difficulties under lognormal assumptions. In all these problems we verify that the variance estimation with isotonic regression is essential for proper mean estimation and beneficial compared to traditional statistical techniques based on local polynomial smoothers.
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Delong, Ł., Szatkowski, M., 2025, One-year and ultimate correlations in dependent claims run-off triangles
Published Paper,
01-02-2024
We investigate bottom-up risk aggregation applied by insurance companies facing reserve risk from multiple lines of business. Since risk capitals should be calculated in different time horizons and calendar years, depending on the regulatory or reporting regime (Solvency II vs IFRS 17), we study correlations of ultimate losses and correlations of one-year losses in future calendar years in lines of business. We consider a multivariate version of a Hertig’s lognormal model and we derive analytical formulas for the ultimate correlation and the one-year correlations in future calendar years. Our main conclusion is that the correlation coefficients that should be used in a bottom-up aggregation formula depend on the time horizon and the future calendar year where the risk emerges. We investigate analytically and numerically properties of the ultimate and the one-year correlations, their possible values observed in practice, and the impact of misspecified correlations on the diversified risk capital.
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