A Shape-Adaptive Gamma-Mixture Power Shanker Distribution: Boundary-Aware Inference, Censored Regression, and Predictive Validation

The Gamma-Mixture Power Shanker distribution models positive lifetimes by allowing the gamma component of a Shanker mixture to change shape and then applying a power transformation. Its three parameters control the latent gamma shape, the rate and mixture weight, and the power transformation. The model contains Power Shanker as an interior submodel and Weibull at the boundary. We develop complete-data and right-censored likelihoods, covariate-dependent rate regression, and likelihood-based diagnostics for choosing among these nested models. A boundary argument establishes continuity, identifies the Weibull nuisance parameters, and verifies positive but small efficient information for the additional shape. A new near-boundary simulation reports bias, empirical standard deviation, root mean squared error, profile-set coverage, and boundary-test rejection rates, revealing appreciable finite-sample uncertainty and limited power. In the 228-patient lung-cancer application, the fitted closure selects Weibull; a multi-start parametric-bootstrap reanalysis supports the boundary reference at the five-percent threshold while showing a discrepancy in its zero mass. Across the complete-data applications, relative fit, evidence against Weibull, evidence against Power Shanker, and absolute goodness of fit lead to different conclusions. The rainfall fit fails two calibrated goodness-of-fit criteria, while the glass-fibre Kolmogorov–Smirnov result is borderline. Repeated predictive validation shows no material GMPS advantage for the lung data. The family therefore provides a testable extension whose additional shape should be retained only when the data support it.