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 statistical 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.