The Unit Folded Normal Distribution: A New Unit Probability Distribution with the Estimation Procedures, Quantile Regression Modeling and Educational Attainment Applications

In this paper, we develop a continuous distribution on the unit interval charac-terized by the distribution of the absolute hyperbolic tangent transformation of a random variable following the normal distribution. The lack of research on the prospect of hyperbolic transformations providing flexible distributions on the unit interval is a motivation for the study. First, we study it theoretically and discuss its properties of interest from a modeling point of view. In particular, it is shown that the proposed distribution accommodates various levels of skewness and kurtosis. Then, some statistical work is performed. We investigate diverse estimation methods for the involved parameters and evaluate their performance through two simulation studies. Subsequently, the quantile regression model derived from the proposed distribution is developed. Two real-world data applications of interest are provided. The first application is about the univariate modeling of the percentage of the educational attainment of some countries, which is one indicator of the edu-cation topic of the Better Life Index (BLI) of the Organization for Economic Co-operation and Development (OECD) countries. The second application is to explain the relationship between the percentage of educational attain-ment of some countries with one indicator of the work-life balance, safety, and health topics of BLI via median quantile regression modeling. For the considered data sets, the proposed distribution and quantile regression mod-els show that they have better modeling abilities than competitive models under some comparison criteria. The results also indicate that covariates are (statistically) significant at any ordinary level of significance for the median response.