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

  • Mustafa Ç. Korkmaz Department of Measurement and Evaluation, Artvin Çoruh University, City Campus, Artvin, 08000, Turkey
  • Christophe Chesneau LMNO, University of Caen Normandie, Caen, 14032, France
  • Zehra Sedef Korkmaz Department of Curriculum and Instruction Program, Artvin Çoruh University, City Campus, Artvin, 08000, Turkey
Keywords: Better life index, unit distribution, hyperbolic tangent function, normal distribution, point estimates, OECD data sets


In this paper, we develop a continuous distribution on the unit interval characterized 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 education 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 attainment 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 models 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.


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Author Biographies

Mustafa Ç. Korkmaz, Department of Measurement and Evaluation, Artvin Çoruh University, City Campus, Artvin, 08000, Turkey

Mustafa Ç. Korkmaz is an Associate Professor in Statistics at Department of Measurement and Evaluation, Education Faculty, Artvin Çoruh University, Artvin, TURKEY. He received his bachelor’s (2008) and master’s (2010) degrees in Statistics from the Selçuk University, TURKEY. He has also received the PhD degree in Statistics from Çukurova University, Adana, TURKEY. His main research interest in the field of two-sided generalized distributions, generalized distribution theory, compound distributions, statistical analysis, regression modeling and quantile regression modeling. He has more than 75 publications in his credit.

Christophe Chesneau, LMNO, University of Caen Normandie, Caen, 14032, France

Christophe Chesneau received the Ph.D. degree in Applied Mathematics from the University of Paris VI in 2006. He has been an Assistant Professor in the Department of Mathematics (LMNO) at the University of Caen-Normandie since 2007. His research activities are focused on applied mathematics, statistics, and probability with applications. Dr. Chesneau has more than 300 international publications to his credit.

Zehra Sedef Korkmaz, Department of Curriculum and Instruction Program, Artvin Çoruh University, City Campus, Artvin, 08000, Turkey

Zehra Sedef Korkmaz is an Assistance Professor in Curriculum Instructions at Department of Educational Sciences, Education Faculty, Artvin Çoruh University, Artvin, TURKEY. She received her bachelor’s (2009), master’s (2012), and PhD (2018) degrees from the Ataturk University, TURKEY. Her main research interest in the field of teacher education, educational statistics, instroctional design, professional development, and statistical analys


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