  In this paper we provide a provably convergent algorithm for the multivariate
Gaussian Maximum Likelihood version of the Behrens--Fisher Problem. Our work
builds upon a formulation of the log-likelihood function proposed by Buot and
Richards \citeBR. Instead of focusing on the first order optimality conditions,
the algorithm aims directly for the maximization of the log-likelihood function
itself to achieve a global solution. Convergence proof and complexity estimates
are provided for the algorithm. Computational experiments illustrate the
applicability of such methods to high-dimensional data. We also discuss how to
extend the proposed methodology to a broader class of problems. We establish a
systematic algebraic relation between the Wald, Likelihood Ratio and Lagrangian
Multiplier Test ($W\geq \mathit{LR}\geq \mathit{LM}$) in the context of the
Behrens--Fisher Problem. Moreover, we use our algorithm to computationally
investigate the finite-sample size and power of the Wald, Likelihood Ratio and
Lagrange Multiplier Tests, which previously were only available through
asymptotic results. The methods developed here are applicable to much higher
dimensional settings than the ones available in the literature. This allows us
to better capture the role of high dimensionality on the actual size and power
of the tests for finite samples.
