An exact frequentist analysis for the Behrens-Fisher problem for solution with general sample sizes
by
Robin Willink
Industrial Research Ltd.

The comparison of the means of two normal distributions with unknown variances is known as the Behrens-Fisher problem. It has an uncomplicated solution if treated as a Bayesian problem with convenience prior distributions, or if treated according to Fisher's concept of fiducial probability. In the late 1940's Welch and Aspin gave a difficult frequentist solution which enabled tabulation of approximate percentage points for sample sizes greater than 6. This talk describes a theoretically-exact frequentist solution which reproduces the results of Welch and Aspin, but is in principle more amenable to calculation with small sample sizes. For fixed sample sizes and a fixed tail-probability the significance curve is a function of the ratio of sample variances. The curve is the solution to an unstable integral equation, but a numerical approximation can be found for good performance down to sample sizes of 3. Unlike the theory of Welch and Aspin, the method does not extend to the case of more than two populations.

Date received: August 27, 2001