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

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Abstract
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.

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