Introducing novel arc cosine-$$\Psi$$ class of distribution with theory and data evaluation related to coronavirus

In recent years, integrating trigonometric techniques into probability models has garnered significant interest. This paper presents a novel trigonometric generator based on the Arc cosine function, referred to as the Arc cos-Ψ distribution. The proposed distribution demonstrates unique and flexible patterns in its probability density function (PDF) and hazard rate function (HRF), showcasing its ability to effectively model both symmetrical and asymmetrical data behaviors. Key mathematical properties of the distribution are thoroughly investigated, including moments, extremum behavior of the PDF and HRF, incomplete moments, quantile function, and entropies. Parameter estimation is carried out using various methods, and their performance is assessed through comprehensive numerical studies. Additionally, a simulation study is conducted to further validate the distribution's properties and estimation techniques. The practical utility and adaptability of the model are demonstrated using two real-world datasets, including COVID-19 data, where the distribution provides an exceptional fit and reveals unique data characteristics. This underscores its potential for modeling complex datasets with intricate structures, making it a valuable addition to the statistical toolkit.