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In this article, we study the consistency of the template estimation with theFr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces isoften used when the observations are deformed or transformed by a group action.We show that in most cases this estimator is actually inconsistent. We exhibita sufficient condition for this inconsistency, which amounts to the folding ofthe distribution of the noisy template when it is projected to the quotientspace. This condition appears to be fulfilled as soon as the support of thenoise is large enough. To quantify this inconsistency we provide lower andupper bounds of the bias as a function of the variability (the noise level).This shows that the consistency bias cannot be neglected when the variabilityincreases.

Template estimation in computational anatomy: Fr\'echet means in top and quotient spaces are not consistent