The unit log-log distribution: a new unit distribution with alternative quantile regression modeling and educational...

Area of Science:

• Statistics
• Econometrics

Background:

• Statistical modeling often requires distributions defined on specific intervals.
• Existing models may not fully capture the complexities of bounded data, particularly in socio-economic contexts.

Purpose of the Study:

• To introduce and analyze the properties of a novel unit log-log distribution.
• To develop and evaluate a new quantile regression model based on this distribution.
• To apply the proposed models to real-world data for educational and socio-economic analysis.

Main Methods:

• Definition and theoretical study of the unit log-log distribution, including its moments and quantile function.
• Development of maximum likelihood estimation for model parameters.
• Construction of a unit quantile regression model and parameter estimation.
• Monte Carlo simulations to assess estimation performance.
• Application to real datasets comparing with existing models.

Main Results:

• The unit log-log distribution exhibits useful properties for modeling data in the (0,1) interval.
• The proposed quantile regression model provides accurate parameter estimations.
• Simulations confirm the effectiveness of the estimation methods.
• Real-data applications show the proposed models outperform competitive alternatives.
• Significant covariates were identified in the analysis of educational measurements and the Better Life Index.

Conclusions:

• The unit log-log distribution and its associated quantile regression model offer a valuable new tool for statistical analysis.
• These models demonstrate enhanced capabilities for explaining relationships in bounded data, such as those found in socio-economic and educational research.
• The findings support the utility of the proposed models for understanding factors influencing national development indicators.