Three Occurrences of the Hyperbolic-Secant Distribution

Although it is the generator distribution of the sixth natural exponential family with quadratic variance function, the Hyperbolic-Secant 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 sta-tistical models. We fill in the gap by showing three examples naturally generating the Hyperbolic-Secant distribution, including Fisher’s analysis of similarity between twins, the Jeffreys ’ prior for contingency tables, and invalid instrumental variables. Key Words: Fisher’s z-transformation; Jeffreys ’ prior; Instrumental variable. 1 The Hyperbolic-Secant Distribution: A Review The Hyperbolic-Secant (HS) distribution is a bell-shaped distribution with mean 0 and variance 1. It can be represented through the standard Cauchy distribution as Y ∼ 2 pi log |C|, (1) where C has the standard Cauchy distribution, and Y has the HS distribution. The density of Y is fY (y) = 1 2