A new generalized alpha power family with applications in healthcare and environmental quality monitoring

This paper introduces a novel class of continuous probability distributions, known as the New Generalized Alpha Power (NGAP) family, designed to improve modeling effectiveness for intricate lifetime and reliability data. The suggested family is formed by applying an exponent transformation to a fundamental cumulative distribution function, enabling the distribution's shape to change dynamically in various probability areas. Key structural and mathematical characteristics of the NGAP family are examined, encompassing closed-form quantile expressions through the Lambert $W_{0}$ function, linear expansions, moments, residual life functions, stress-strength reliability, order statistics, Rényi entropy, and extropy measures. Statistical inference for the unknown parameter vector $\Theta = (\alpha, \beta, \theta, k)^{T}$ is performed utilizing fifteen non-Bayesian estimation techniques—including Maximum Likelihood Estimation (MLE), distance-oriented methods, and spacing-based approaches. Monte Carlo simulations assess how well these estimators perform with sample sizes varying from $n = 10$ to $n = 500$. Numerical results indicate that as the sample size $n$ grows, the absolute bias and mean squared error (MSE) consistently decline across parameters; for instance, with $\alpha = 1.25, \beta = 0.75, \theta = 1.50, k = 1.75$, the MLE estimation for $\alpha$ shows an absolute bias (and MSE) dropping from $6.3235$ ($679.9716$) at $n = 10$ to $0.0873$ ($0.2146$) at $n = 200$ and $0.0564$ ($0.1339$) at $n = 300$, whereas the Kolmogorov Estimator (KE) reaches a bias of $0.0125$ ($0.0035$) at $n = 200$. In general, the Maximum Product of Spacings Estimator (MPSE) reliably secures the highest rankings in various simulation scenarios. A specific variant centered on the Weibull baseline (NGAP-Weibull) is analyzed thoroughly, along with submodels like the NGAP-Exponential and NGAP-Rayleigh distributions. Applications to actual datasets show that the NGAP model achieves better goodness-of-fit than current competing distributions, serving as a flexible tool with real-world uses in reliability engineering, biomedical studies, and survival analysis.