A novel Weighted Exponential-Lindley distribution with Bayesian estimation under progressive Type-II censoring and a Bayesian-Calibrated Variational Autoencoder for survival prediction

One-parameter lifetime models, such as the exponential and Lindley distributions, and two-parameter extensions such as the weighted Lindley distribution cannot separately control tail weight and dispersion, since a single shape parameter governs both. We address this with the Weighted Exponential-Lindley (WEL) distribution, a three-parameter model built by applying a length-biased weight to a mixture of exponential and Lindley components, with closed-form density, survival function, and an exact two-component gamma mixture representation. Using this representation and Glaser's failure-rate criterion, we prove that the WEL hazard rate is strictly increasing on its entire parameter space, with the scale parameter as its asymptotic level; the weight parameter then gives independent, tunable control over dispersion, skewness, and kurtosis are unavailable in the Lindley or weighted Lindley families. We develop maximum likelihood, method-of-moments, and Bayesian estimation under progressive Type-II censoring, validated through a Monte Carlo study across several parameter configurations and sample sizes, reporting bias, MSE, interval coverage, computation time, and convergence rates, including non-convergent method-of-moments runs rather than only successful ones. We then propose the Bayesian-Calibrated Variational Autoencoder Survival Model (BC-VAESM), whose decoder outputs individual, covariate-dependent WEL parameters, trained end-to-end via a censored negative log-likelihood. We evaluate both contributions on real, publicly available survival datasets, including censored clinical trial data and complete reliability/biological benchmarks, one chosen specifically to illustrate WEL's inability to represent bathtub hazards and one matching its increasing-hazard property. Fitting the censored datasets under an explicit right-censored likelihood, distinguished from the progressive Type-II scheme used in the Bayesian machinery, WEL is compared against an expanded set of competing one-, two-, and three-parameter lifetime models, including further weighted distributions. WEL is the best-fitting model on one dataset but is outperformed on others, and we report all outcomes without selective emphasis. In five-fold cross-validation, the BC-VAESM modestly exceeds a penalized Cox model and an otherwise identical Weibull-decoder autoencoder, though the improvement is not statistically significant at the sample size examined. Every estimate reported here comes from real, reproducible computation on real data. MSC 2020 62N05; 62F15; 62E15; 68T07; 62P10.