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

By Alexandre Belloni, Gustavo Didier. 2008

28 citations recorded by the source corpus.

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 [5]. 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 ≥ LR ≥ 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.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

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Citations in the corpus

Some solutions to the multivariate Behrens–Fisher problem for dissimilarity‐based analyses (2017, background): “…to the multivariate BFP (e.g., Johnson & Weerhandi 1988; Coombs & Algina 1996; Christensen & Rencher 1997; Gamage, Mathew & Weerhandi 2004; Belloni & Didier 2008; Krishnamoorthy & Lu 2010) assume variables to be multivariate normal (MVN), and are not amenable to high-dimensional problems…”
A parametric bootstrap solution to the MANOVA under heteroscedasticity (2010, result): “Recent comparison studies by Krishnamoorthy and Yu 18 , Hirokazu and Ke-Hai 20 , Park and Sinha 19 , and Belloni and Didier 21 indicate that the modified Nel and Van der Merwe's (MNV) test proposed in Krishnamoorthy and Yu 18 is comparable to, or better than, other affine invariant tests.”