Unit Rayleigh Half-Normal Distribution: Bayesian and Non-Bayesian Inference, Regression Model for Bounded Response Data and Application

This paper introduces a novel one-parameter bounded distribution, called the Unit Rayleigh Half-Normal (URHN) distribution, designed for modeling data on the unit interval (0,1), which frequently arises in many fields such as economics, actuarial science, and medicine. We examine the structural properties of the URHN distribution and derive its key statistical and reliability characteristics. Parameter estimation is carried out using a Bayesian method via Markov Chain Monte Carlo (MCMC), along with other standard techniques, including maximum likelihood, maximum product spacing, and least squares methods. The performance of these estimation methods is evaluated through simulation studies. To broaden the scope of the URHN distribution, we introduce a regression model by reparameterizing it in terms of its mean, enabling the incorporation of covariates for modeling bounded response variables. The practical utility of the proposed model is demonstrated using data representing the percentage of educational attainment in countries of the Organization for Economic Co-operation and Development (OECD), which is one of the indicators within the education dimension of the Better Life Index (BLI). We examine the relationship between educational attainment and three explanatory indicators: homicide rate, housing expenditure, and labor market insecurity. The results indicate that only labor market insecurity has a statistically significant effect on educational attainment. With its theoretical foundation, modeling flexibility, and empirical effectiveness, the URHN model emerges as a competitive alternative to established bounded distributions.