The t family and their close and distant relations

Student's t distribution arose as a normal-based sampling distribution and has huge application as such. In modern times, the t distribution has also found considerable use as a symmetric heavy-tailed distribution for empirical data modelling. In this the centenary year of Student's introduction of the t distribution, this paper constitutes a personal-and sometimes somewhat tangential-tribute to the Student-t family of distributions, by way of an exploration of some of its close and not-so-close relations. I start with a brief reminder of the (symmetric) Student-t family, with particular focus on the t distribution on two degrees of freedom which turns out to play an important role in what follows. The “close relations” of the t distribution include a number of 'skew-t' distributions, of which I shall briefly introduce and comment on three. The “distant relations” of the t distribution are a further trio of four-parameter families of distributions allowing control of skewness and tailweights, with particular emphasis on members of the families with (t-like) 'power' density tails.