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

This paper introduces the New Generalized Alpha Power (NGAP) family of continuous distributions, which utilizes an exponent transformation to enhance modeling flexibility for complex lifetime data, and validates its superior performance through comprehensive mathematical analysis, extensive simulation of fifteen estimation methods, and successful applications in healthcare and environmental quality monitoring.

Original authors: Shamshad Ur Rasool, Abdoulie Faal, Okechukwu J. Obulezi

Published 2026-09-23

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CC BY 4.0

Original authors: Shamshad Ur Rasool, Abdoulie Faal, Okechukwu J. Obulezi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of reliability engineering and survival analysis, scientists and engineers constantly ask a simple but vital question: how long will something last? Whether it is a medical implant, a bridge component, or a patient's survival time after a diagnosis, the answer is rarely a single number. Instead, the data is a story of risk that changes over time. Sometimes the risk of failure rises steadily as a machine ages; other times, it starts high, drops, and then rises again, or perhaps it stays flat. To make sense of these complex patterns, researchers use mathematical models called probability distributions. These are not physical objects but flexible shapes that describe how likely different outcomes are. The goal is to find a shape that fits the messy reality of the data perfectly, allowing experts to predict future behavior with confidence. For decades, a model known as the Weibull distribution has been a standard tool for this work because it is simple and adaptable, but real-world data often contains twists and turns that even this flexible tool cannot capture without becoming overly complicated.

A team of researchers has now introduced a new mathematical framework designed to handle these intricate patterns with greater ease and precision. They call this the New Generalized Alpha Power family. Imagine taking a standard, reliable shape and adding a special mechanism that allows it to stretch, shrink, and twist in specific ways to match the data exactly, without needing to add a confusing array of extra rules. This new family of models is built by applying a specific transformation to the baseline shape, which gives it the power to adapt to different types of skewness and tail behavior. The researchers focused heavily on a specific version of this new family that is based on the classic Weibull distribution. By combining the reliability of the old model with the flexibility of the new transformation, they created a tool that can describe everything from steadily increasing risks to complex, unimodal patterns where the danger peaks and then fades.

The core of this work involves proving that this new mathematical structure is sound and that its parameters can be uniquely identified. The researchers showed that if two different sets of settings produce the exact same curve, those settings must actually be the same, ensuring the model is stable and reliable for statistical use. They then derived a series of mathematical properties that describe how this new distribution behaves, including how to calculate its average values, its remaining life expectancy, and how it handles stress and strength. Crucially, they developed a way to generate random numbers that follow this specific distribution, which is essential for testing the model's performance before applying it to real data.

To see if this new tool actually works better than existing ones, the team conducted a massive series of computer simulations. They generated thousands of fake datasets of varying sizes, ranging from just ten observations up to five hundred, and tried to fit the new model to them using fifteen different estimation techniques. These techniques are like different methods for finding the best fit, ranging from standard maximum likelihood estimation to more specialized distance-based approaches. The results of these simulations were clear: as the amount of data increased, the accuracy of the estimates improved significantly. The researchers found that one particular method, known as the maximum product of spacings estimator, consistently outperformed the others, providing the most accurate and stable results across almost every scenario they tested. Other methods, including the standard maximum likelihood approach, also performed well, but the new estimator secured the top rankings in the majority of their tests.

The paper then moved from simulated data to the real world, applying the new model to actual datasets from healthcare and environmental monitoring. In these applications, the new model was compared against a variety of established competing distributions. The results showed that the new generalized model provided a better fit to the real-world data than the older models. This means it could describe the underlying patterns of failure or survival more accurately, capturing nuances that the previous tools missed. The researchers demonstrated that this new family is not just a theoretical exercise but a practical tool that can be used to improve predictions in fields where understanding the timing of events is critical. By offering a more flexible and mathematically manageable way to model complex lifetime data, this work provides engineers and scientists with a sharper instrument for analyzing risk and reliability in an unpredictable world.

Technical Summary: A New Generalized Alpha Power Family with Applications in Healthcare and Environmental Quality Monitoring

Problem Statement
The analysis of lifetime and reliability data is critical in fields such as reliability engineering, biomedical studies, and survival analysis. However, real-world data often exhibit complex characteristics—including varying skewness, significant tail weights, and non-monotone hazard-rate patterns (e.g., bathtub-shaped or unimodal)—that traditional lifetime models, such as the standard Weibull distribution, fail to capture effectively. While numerous generator-based families have been developed to address these limitations by adding shape parameters, many current approaches introduce excessive structural complexity. This complexity often hinders mathematical derivations, likelihood inference, numerical optimization, and the interpretability of parameters. Consequently, there is a need for generator mechanisms that balance substantial flexibility with mathematical and computational manageability.

Methodology
To address these gaps, the authors propose a novel class of continuous probability distributions termed the New Generalized Alpha Power (NGAP) family. The NGAP framework is constructed by applying an exponent transformation to a baseline cumulative distribution function (CDF), $F(x; \Omega)$ . Unlike traditional power-generated models that utilize constant exponents, the NGAP transformation allows the effect to flexibly adjust across various probability areas.

The CDF and Probability Density Function (PDF) of the NGAP family are defined as:
$G(x; \alpha, \beta) = \{F(x; \Omega)\}^\alpha e^{-\beta F(x; \Omega)}$
$g(x; \alpha, \beta, \Omega) = \{F(x; \Omega)\}^\alpha e^{-\beta F(x; \Omega)} \alpha f(x; \Omega) e^{-\beta F(x; \Omega)} \left[ \frac{1}{F(x; \Omega)} - \beta \ln F(x; \Omega) \right]$
where $\alpha$ and $\beta$ are shape parameters governing skewness, kurtosis, and tail behavior, and $\Omega$ represents the parameters of the baseline distribution.

The paper focuses specifically on the NGAP-Weibull (NGAPW) distribution, where the Weibull distribution serves as the baseline. The study establishes the identifiability of the parameter vector $\Theta = (\alpha, \beta, \theta, k)^T$ , proving that distinct parameter vectors yield distinct cumulative distribution functions.

Key Contributions and Structural Properties
The paper provides a rigorous mathematical treatment of the NGAPW model, including:

Closed-Form Quantile Functions: Derivation of the quantile function using the principal branch of the Lambert $W_0$ function, facilitating random number generation.
Linear Mixture Expansions: Explicit series representations for the CDF and PDF as infinite linear combinations of exponentiated Weibull densities. This facilitates the derivation of:
- Raw and incomplete moments.
- Order statistics.
- Stress-strength reliability metrics ( $R = P(X_1 > X_2)$ ).
- Entropy measures, including Rényi entropy and Tsallis extropy.
- Mean residual life functions.
Submodels: Identification of special cases, including the NGAP-Exponential and NGAP-Rayleigh distributions, demonstrating the model's unification of various lifetime distributions.
Hazard Rate Flexibility: Analysis showing the NGAPW hazard function can assume monotone increasing, monotone decreasing, unimodal (upside-down bathtub), and bathtub shapes depending on parameter configurations.

Statistical Inference and Simulation
The study evaluates the performance of fifteen distinct non-Bayesian estimation techniques for the parameter vector $\Theta$ . These include:

Maximum Likelihood Estimation (MLE).
Distance-oriented methods (Anderson-Darling, Cramér-von Mises, Kolmogorov, and tail-specific variants).
Spacing-based approaches (Maximum Product of Spacings, Minimum Spacing Absolute Distance, etc.).

A comprehensive Monte Carlo simulation was conducted with sample sizes ranging from $n=10$ to $n=500$ . The performance of estimators was assessed using absolute bias, Mean Squared Error (MSE), Mean Relative Error (MRE), and various distance metrics.

Results: The simulations indicate that as sample size increases, the absolute bias and MSE consistently decline across all 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 $n = 300$ .
Ranking: The Maximum Product of Spacings Estimator (MPSE) consistently secured the highest rankings across various simulation scenarios. The Maximum Likelihood Estimator (MLE) and other spacing/least-squares methods provided competitive performance under specific parameter choices. Notably, the Kolmogorov Estimator (KE) reaches a bias of 0.0125 (0.0035) at $n = 200$ .

Empirical Applications
The NGAPW model was applied to real-world datasets from healthcare (biomedical) and environmental quality monitoring sectors.

Goodness-of-Fit: The NGAPW model demonstrated superior goodness-of-fit compared to established competing distributions and baseline generalizations.
Utility: The results confirm the model's effectiveness in fitting heavy-tailed and non-monotone hazard rate dynamics, validating its utility as a flexible tool for reliability engineering, biomedical studies, and survival analysis.

Significance
The paper claims that the NGAP family offers a unified modeling framework that combines the flexibility required to model complex real-world data behaviors with the mathematical tractability necessary for practical inference. By avoiding the excessive complexity found in some existing generated families, the NGAPW distribution provides a robust alternative for researchers and practitioners dealing with intricate lifetime data, offering improved modeling effectiveness without sacrificing computational feasibility.

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