A Gamma–Lindley q-distribution for deep probabilistic regression of positive data

We introduce a mathematically consistent Gamma–Lindley q -distribution and develop a support-aware deep probabilistic regression framework for positive-valued responses. The proposed model is constructed within Jackson q -calculus and uses the q -gamma function to obtain an explicit normalization mechanism together with closed-form expressions for raw moments, the conditional mean, and the conditional variance. We establish positivity, exact normalization, and convergence to the classical admissible Gamma–Lindley distribution as \(q → 1^-\) . Building on these analytic results, we embed the proposed distribution into a neural conditional density estimator whose parameters depend on the covariates through constraint-preserving output maps. In particular, the model learns the conditional support endpoint explicitly, which guarantees likelihood feasibility and stabilizes training under finite q -support. Numerical experiments validate the q -gamma Jackson-integral representation, normalization and moment identities, sampling on the Jackson grid, and the classical-limit behavior. In synthetic probabilistic-regression experiments, the identifiable version of the proposed model accurately recovers the latent parameter functions while providing sharper predictive intervals and substantially improved conditional-mean estimation compared with a baseline likelihood-only model. We further benchmark the proposed estimator against Gamma, Weibull, log-normal and generalized-gamma deep distributional-regression baselines on both synthetic data and a real positive-valued dataset, where it attains competitive predictive accuracy together with well-covered predictive intervals. These results show that the Gamma–Lindley q -distribution provides a theoretically grounded and practically effective foundation for deep probabilistic regression of positive data.