Multiple solutions to the likelihood equations in the Behrens-Fisher problem

TL;DR

This paper investigates the likelihood function's multiple solutions in the Behrens-Fisher problem, showing that multimodality contradicts the null hypothesis as sample sizes grow, with finite-sample bounds and asymptotic results.

Contribution

It establishes that likelihood multimodality under the null hypothesis diminishes with increasing sample sizes and provides bounds and asymptotic analysis for this behavior.

Findings

Likelihood function can have multiple local maxima.

Multimodality probability tends to zero under the null as sample sizes increase.

Finite-sample bounds on the probability of multimodality.

Abstract

The Behrens-Fisher problem concerns testing the equality of the means of two normal populations with possibly different variances. The null hypothesis in this problem induces a statistical model for which the likelihood function may have more than one local maximum. We show that such multimodality contradicts the null hypothesis in the sense that if this hypothesis is true then the probability of multimodality converges to zero when both sample sizes tend to infinity. Additional results include a finite-sample bound on the probability of multimodality under the null and asymptotics for the probability of multimodality under the alternative.