Exponentiated Stereographic Semicircular Family Distribution

Abstract

In this paper, the exponential semi-circular solid-state projection family (ESD) is presented as a basis for transforming linear probability distributions into highly flexible and accurate distributions for representing circular statistical data. The exponential family rule and the solid-state projection rule are used to generate semi-circular distributions. A probability density function is derived and applied to the linear Fréchet distribution to obtain a new circular probability distribution characterized by accuracy, comprehensiveness, and flexibility in representing circular data. Some mathematical properties of the proposed distribution are derived, and estimators of the unknown parameters of the proposed distribution are derived using the Maximum Likelihood (MLE) method. The study was applied to a sample size of 100 observations representing the angles of posterior corneal curvature of the eye. It was concluded that the proposed distribution is more efficient in representing circular data compared to the semi-circular solid-state Pareto-Weibel model and the semi-circular solid-state gamma distribution, based on the Akaike criterion (AIC), the Bayes-Akaike test (BIC), and the consistent Akaike test (CAIC), After the proposed distribution passes the real-data fit test according to the chi-square test.