Shrinkage-Gating Zero-Inflated Tweedie Boosting with Exact Degeneracy: Boundary Selection, Profile Likelihood Estimation, and Actuarial Calibration

Abstract The Tweedie compound Poisson–Gamma distribution possesses a natural zerogenerating mechanism. Introducing a structural zero component into insurance claim data may therefore improve zero-probability calibration, yet it may also mistakenly attribute dispersion misspecification or sampling variability to excess zero inflation. The standard logit parameterization suffers from a neglected technical impediment: the absence of zero inflation corresponds to the logit predictor tending to negative infinity, and conventional smooth penalties only drive the structural zero probability toward zero without recovering the standard Tweedie model at any finite parameter value. This paper proposes a shrinkage-gating zero-inflated Tweedie (SG-ZIT) boosting model, specified as ωi = 1 − exp(−τ ri), where τ ≥ 0 denotes the global gating intensity, and the positive function ri characterizes covariate heterogeneity and is normalized to unit mean on the training sample. The boundary case τ = 0 corresponds exactly to the standard Tweedie model at a finite parameter point. We fix the Tweedie component and the gating shape on an independent data fold, and then estimate the gating intensity via profile likelihood with a λτ penalty. For fixed nuisance parameters, the profile objective reduces to a one-dimensional convex function. We derive the first and second derivatives for zero and positive observations, the necessary and sufficient KKT conditions for exact degeneracy, the monotonicity of the shrinkage path, a characterization of the non-convexity of the joint likelihood, as well as finite-sample bounds on the false-selection probability and asymptotic consistency of boundary selection under the null hypothesis τ = 0. The algorithm updates the Tweedie component using posterior membership weights on the training fold, estimates τ along each path on a calibration fold, selects the penalty parameter via the one-standard-error rule on a selection fold, and evaluates the test fold exactly once. Simulation studies examine scenarios with and without genuine structural zeros using independent replicates. The real-data application uses the publicly available freMTPL2 dataset, a French motor third-party liability portfolio. The results demonstrate that SG-ZIT reduces the Brier score and log loss for zero probabilities when structural zeros are present. In simulations without structural zeros, the selected gating mechanism yields τ = 0 exactly, rather than merely producing numerically negligible zero-inflation probabilities. The freMTPL2 results illustrate that an extremely high raw zero proportion does not automatically constitute evidence of additional zero inflation. The primary contribution of this paper is to provide a transparent and verifiable complexity path: the extra zero component is retained only when supported by independent out-of-sample scoring; otherwise, the model degenerates strictly to the standard Tweedie.