Three Occurrences of the Hyperbolic-Secant Distribution

Abstract

Although it is the generator distribution of the sixth natural exponential family with quadratic variance function, the Hyperbolic-Secant (HS) distribution is much less known than other distributions in the exponential families. Its lack of familiarity is due to its isolation from many widely used statistical models. We fill in the gap by showing three examples naturally generating the HS distribution, including Fisher’s analysis of similarity between twins, the Jeffreys’ prior for contingency tables, and invalid instrumental variables.

• Fisher’s z-transformation
• Instrumental variable
• Jeffreys’ prior