A novel three-parameter Gompertz-Lomax Distribution with a physics-informed survival network: statistical theory, hazard shape characterization, and real-world applications.

This article introduces the Gompertz-Lomax Distribution (GLD), a novel three-parameter continuous lifetime model whose composite hazard rate additively combines an increasing Gompertz component with a decreasing Lomax component. The resulting closed-form survival function nests the Gompertz and Lomax distributions as exact special cases and accommodates increasing, decreasing, and bathtub-shaped hazard profiles within a single parametric framework. Complete closed-form expressions are derived for the raw and central moments, incomplete moments, quantile function, mean residual life, stress-strength reliability, stochastic ordering, Rényi entropy, and Bonferroni-Lorenz inequality curves. Four estimation procedures, namely maximum likelihood, method of moments, least squares, and weighted least squares, are developed and validated through a Monte Carlo simulation study comprising N = 10, 000 replications, with finite-sample performance compared across all four estimators. As the primary machine-learning contribution, a Physics-Informed Survival Network (PISN) is introduced: a deep neural network that maps subject-level covariates to individual GLD shape and scale parameters while encoding the theoretically derived hazard shape condition as a smooth, differentiable penalty in the training loss. Empirical evaluation on three benchmark datasets, namely, leukemia patient survival, COVID-19 hospital survival, and electronic component failure times, demonstrates that the GLD achieves among the lowest Kolmogorov-Smirnov statistics and the most favorable Akaike Information Criterion (AIC), corrected AIC (AICC), and Bayesian Information Criterion (BIC) on all three datasets, with ΔAIC exceeding five units over the nearest competitor on the leukemia dataset and ΔAIC = 30.6 on the electronic component dataset, which exhibits a bathtub-shaped hazard rate.