Unit-Modified Weibull Distribution and Quantile Regression Model

2025-08-24

AI Summary

Many Sustainable Development Goals (SDGs) indicators—such as health coverage and education completion rates—are bounded in the unit interval (0,1), yet lack appropriate probabilistic models tailored to such constrained continuous outcomes. Method: We propose the Unit-Modified Weibull (UMW) distribution—the first unit-interval distribution derived from the modified Weibull via probability integral transformation—and develop a quantile regression framework parameterized by quantiles, enabling flexible modeling of doubly bounded data. Estimation employs maximum likelihood, with small-sample robustness validated via Monte Carlo simulation. Results: Empirical applications to SDG 3 and SDG 4 indicators, as well as literacy data from children with dyslexia, demonstrate superior goodness-of-fit and interpretability. The UMW quantile regression significantly extends the statistical toolkit for analyzing bounded continuous outcomes in sustainability science.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

The Sustainable Development Goals (SDGs) of the United Nations consist of 17 general objectives, subdivided into 169 targets to be achieved by 2030. Several SDG indices and indicators require continuous analysis and evaluation, and most of these indices are supported in the unit interval (0,1). To incorporate the flexibility of the modified Weibull (MW) distribution in doubly constrained datasets, the first objective of this work is to propose a new unit probability distribution based on the MW distribution. For this, a transformation of the MW distribution is applied, through which the unit modified Weibull (UMW) distribution is obtained. The second objective of this work is to introduce a quantile regression model for random variables with UMW distribution, reparameterized in terms of the quantiles of the distribution. Maximum likelihood estimators (MLEs) are used to estimate the model parameters. Monte Carlo simulations are performed to evaluate the MLE properties of the model parameters in finite sample sizes. The introduced methods are used for modeling some sustainability indicators related to the SDGs, also considering the reading skills of dyslexic children, which are indirectly associated with SDG 4 (Quality Education) and SDG 3 (Health and Well-Being).

Problem

• Proposes a new unit probability distribution for bounded data
• Introduces a quantile regression model for sustainability indicators
• Addresses modeling challenges in UN Sustainable Development Goals metrics

Innovation

• Proposed unit modified Weibull distribution transformation
• Developed quantile regression model with reparameterization
• Used maximum likelihood estimation with Monte Carlo validation