A study on the skew-normal distribution with application to the development of false discovery rate control methods

In recent years, the skew-normal distribution, developed as an extension of the classical normal distribution, has garnered significant attention due to its capability to model asymmetrical data often encountered in real-world applications. By introducing a skewness parameter, this distribution retains many desirable properties of the normal distribution while offering enhanced flexibility, making it a valuable tool for statistical modeling. This paper provides a comprehensive review of the theoretical developments and applications of the skew-normal distribution. We systematically summarize the fundamental distributional properties including marginalization, closure under linear transformations, and quadratic forms. Moreover, several construction mechanisms leading to skew-normal distributions are discussed, such as conditional selection, hidden truncation, convolution-based representation, and selective reporting. These structural insights provide a unified understanding of how skewness arises within this family of distributions and lay the foundation for the development of false discovery rate control methods.