Abstract

We develop a unified likelihood-based framework for limited outcomes built on the two-piece normal family. The framework includes a censored specification that accommodates boundary inflation, a doubly truncated specification on (0,1) for rates and proportions, and a survival formulation with a log-two-piece normal baseline and Gamma frailty to account for unobserved heterogeneity. We derive closed-form building blocks (pdf, cdf, survival, hazard, and cumulative hazard), full log-likelihoods with score functions and observed information, and stable reparameterizations that enable routine optimization. Monte Carlo experiments show a small bias and declining RMSE with increasing sample size; censoring primarily inflates the variability of regression coefficients; the scale parameter remains comparatively stable, and the shape parameter is most sensitive under heavy censoring. Applications to HIV-1 RNA with a detection limit, household food expenditure on (0,1), labor-supply hours with a corner solution, and childhood cancer times to hospitalization demonstrate improved fit over Gaussian, skew-normal, and beta benchmarks according to AIC/BIC/CAIC and goodness-of-fit diagnostics, with model-implied censoring closely matching the observed fraction. The proposed formulations are tractable, flexible, and readily implementable with standard software.