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

In this paper, we propose a new distribution, named unit log-log distribution, defined on the bounded (0,1) interval. Basic distributional properties such as model shapes, stochastic ordering, quantile function, moments, and order statistics of the newly defined unit distribution are studied. The maximum likelihood estimation method has been pointed out to estimate its model parameters. The new quantile regression model based on the proposed distribution is introduced and it has been derived estimations of its model parameters also. The Monte Carlo simulation studies have been given to see the performance of the estimation method based on the new unit distribution and its regression modeling. Applications of the newly defined distribution and its quantile regression model to real data sets show that the proposed models have better modeling abilities than competitive models. The proposed unit quantile regression model has targeted to explain linear relation between educational measurements of both OECD (Organization for Economic Co-operation and Development) countries and some non-members of OECD countries, and their Better Life Index. The existence of the significant covariates has been seen on the real data applications for the unit median response.

A new alternative quantile regression model for the bounded response with educational measurements applications of OECD countries.

This article introduces a new distribution with two tuning parameters specified on the unit interval. It follows from a 'hyperbolic secant transformation' of a random variable following the Weibull distribution. The lack of research on the prospect of hyperbolic transformations providing flexible distributions over the unit interval is a motivation for the study. The main distributional structural properties of the new distribution are established. The different estimation methods and two simulation works have been derived for model parameters. Subsequently, we develop a related quantile regression model for further statistical perspectives. We consider two real data applications based on the educational measurements of both OECD and some non-members of OECD countries. Our regression model aims to relate the desire to get top grades on certain young students in the OECD countries with some of their Education and School Life Index such as reading performance, work environment at home, and paid work experience. It is shown that the elaborated quantile regression model has a better fitting power than famous regression models when the unit response variable possesses skewed distribution as well as two independent variables are significant in the statistical sense at any standard significance level for the median response.