This study presents a new three-parameter beta distribution defined on the unit interval, which can have increasing, decreasing, left-skewed, right-skewed, approximately symmetric, bathtub, and upside-down bathtub shaped densities, and increasing, U, and bathtub shaped hazard rates. This model can define well-known distributions with various parameters and supports, such as Kumaraswamy, beta exponential, exponential, exponentiated exponential, uniform, the generalized beta of the first kind, and beta power distributions. We present a comprehensive account of the mathematical features of the new model. Maximum likelihood methods and a Bayesian method under squared error and linear exponential loss functions are presented; also, approximate confidence intervals are obtained. We present a simulation study to compare all the results. Two real-world data sets are analyzed to demonstrate the utility and adaptability of the proposed model.

Reliability studies frequently employ progressive censoring schemes that remove surviving units during testing, yet statistical inference under such designs remains vulnerable to parametric model misspecification. When distributional assumptions fail, conventional maximum likelihood estimators converge to systematically biased limits, producing confidence intervals with severely degraded coverage. We develop a flexible inferential framework that models the hazard function through a neural network architecture, avoiding commitment to a parametric family. To quantify uncertainty, we introduce a stratified weighted bootstrap procedure that preserves the dependency structure induced by progressive removals. We establish that the proposed estimator achieves the minimax optimal nonparametric rate $n^{-\alpha /(2\alpha +1)}$ for $\alpha$-smooth hazard functions and prove that the bootstrap consistently approximates the sampling distribution, yielding asymptotically valid pointwise confidence intervals for the survival function. A local asymptotic analysis precisely characterizes the efficiency–robustness tradeoff. Comprehensive simulations comparing against parametric methods, penalized splines, piecewise exponential models, and kernel estimators demonstrate that our method maintains 92–94% coverage under misspecification, whereas parametric alternatives collapse to 40–45% and simpler nonparametric methods achieve only 85–91%. The neural network architecture provides 23–29% lower integrated mean squared error than penalized splines using the same bootstrap, confirming that both components of our framework contribute to performance. Computational requirements remain practical: parallelized bootstrap inference completes in under 25 s on an 8-core processor for typical sample sizes. Application to electronic component lifetime data illustrates how the methodology yields materially different reliability assessments with direct implications for warranty planning.