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In this paper we provide a provably convergent algorithm for the multivariateGaussian Maximum Likelihood version of the Behrens--Fisher Problem. Our workbuilds upon a formulation of the log-likelihood function proposed by Buot andRichards \citeBR. Instead of focusing on the first order optimality conditions,the algorithm aims directly for the maximization of the log-likelihood functionitself to achieve a global solution. Convergence proof and complexity estimatesare provided for the algorithm. Computational experiments illustrate theapplicability of such methods to high-dimensional data. We also discuss how toextend the proposed methodology to a broader class of problems. We establish asystematic algebraic relation between the Wald, Likelihood Ratio and LagrangianMultiplier Test ($W\geq \mathit{LR}\geq \mathit{LM}$) in the context of theBehrens--Fisher Problem. Moreover, we use our algorithm to computationallyinvestigate the finite-sample size and power of the Wald, Likelihood Ratio andLagrange Multiplier Tests, which previously were only available throughasymptotic results. The methods developed here are applicable to much higherdimensional settings than the ones available in the literature. This allows usto better capture the role of high dimensionality on the actual size and powerof the tests for finite samples.

On the Behrens--Fisher problem: A globally convergent algorithm and a finite-sample study of the Wald, LR and LM Tests