Unit-Modified Weibull Distribution and Quantile Regression Model

TL;DR

This paper introduces a new unit probability distribution based on the modified Weibull distribution and develops a quantile regression model for it, applied to sustainability indicators and educational data.

Contribution

It proposes the unit modified Weibull distribution and a corresponding quantile regression model, extending the analysis of bounded data in sustainability and education contexts.

Findings

Monte Carlo simulations validate the MLE estimators.

The model effectively fits SDG-related indicators.

Application to dyslexia data demonstrates practical utility.

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).