A novel stereographic semi-circular distribution and enhanced analysis of the posterior corneal curvature of eye

Circular distributions within the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mo>-</mml:mo> <mml:mi>π</mml:mi> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math> radians range describe two-dimensional directions by mapping points onto a unit circle. These distributions are vital in diverse fields such as medicine, ecology, and environmental studies, where measurements are expressed in terms of angles. However, when these distributions involve measuring angles within the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math> radians range, they constitute axial or semi-circular data instead of circular data. This research seeks to introduce the semi-circular Marshall-Olkin extended Burr-XII distribution tailored for semi-circle datasets. Objectives encompass presenting its fundamental characteristics and applications. The inverse stereographic projection technique is applied for its development, deriving characteristics like trigonometric moments, mode, hazard function, and survival function. Five estimation techniques assess the distribution's parameters. Monte Carlo simulations evaluate parameter estimation methods for different sample sizes. Modeling the semi-circular Marshall-Olkin extended Burr-XII distribution with real-life semi-circle data of posterior corneal curvature of eye demonstrates its adaptability. Comparisons with existing distributions affirm its effectiveness. Extending to the <i>l</i>-axial model produces the Stereographic-<i>l</i>-axial Marshall-Olkin extended Burr-XII distribution, offering a distinct probability density function (pdf). This transformation gives rise to specific scenarios and new models. The proposed semi-circular Marshall-Olkin extended Burr-XII distribution proves adept at handling real-world semi-circular data. The extension to the <i>l</i>-axial model and subsequent transformations introduces innovative models, demonstrated by superior compatibility in both circular and semi-circular datasets.