A.18 Behrens’ distribution

X is said to have Behrens’ (or Behrens–Fisher or Fisher–Behrens) distribution with degrees of freedom and and angle , denoted

if X has the same distribution as

where T1 and T2 are independent and

Equivalently, X has density

where

over the whole real line, where

This distribution naturally arises as the posterior distribution of

when we have samples of size from and of size from and neither nor is known, and conventional priors are adopted. In this case, in a fairly obvious notation

An approximation to this distribution due to Patil (1965) is as follows.

Define

Then, approximately,

Obviously b is usually not an integer, and consequently this approximation requires interpolation in the t tables.

Clearly Behrens’ distribution has mean and variance

using the mean and variance of t distributions and the independence of T1 and T2. The distribution is symmetrical and unimodal and hence the mean, mode and median are all equal, so