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Compositional data consists of vectors of proportions whose components sum to1. Such vectors lie in the standard simplex, which is a manifold with boundary.One issue that has been rather controversial within the field of compositionaldata analysis is the choice of metric on the simplex. One popular possibilityhas been to use the metric implied by logtransforming the data, as proposed byAitchison [1, 2]; and another popular approach has been to use the standardEuclidean metric inherited from the ambient space. Tsagris et al. [21] proposeda one-parameter family of power transformations, the $\alpha$-transformations,which include both the metric implied by Aitchison's transformation and theEuclidean metric as particular cases. Our underlying philosophy is that, withmany datasets, it may make sense to use the data to help us determine asuitable metric. A related possibility is to apply the $\alpha$-transformationsto a parametric family of distributions, and then estimate a along with theother parameters. However, as we shall see, when one follows this last approachwith the Dirichlet family, some care is needed in a certain limiting case whicharises $(\alpha \neq 0)$, as we found out when fitting this model to real andsimulated data. Specifically, when the maximum likelihood estimator of a isclose to 0, the other parameters tend to be large. The main purpose of thepaper is to study this limiting case both theoretically and numerically and toprovide insight into these numerical findings.

Gaussian asymptotic limits for the $\alpha$-transformation in the analysis of compositional data