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The Central Limit Theorem (CLT) for extrinsic and intrinsic means onmanifolds is extended to a generalization of Fr\'echet means. Examples are theProcrustes mean for 3D Kendall shapes as well as a mean introduced by Ziezold.This allows for one-sample tests previously not possible, and to numericallyassess the `inconsistency of the Procrustes mean' for a perturbation model and`inconsistency' within a model recently proposed for diffusion tensor imaging.Also it is shown that the CLT can be extended to mildly rank deficientdiffusion tensors. An application to forestry gives the temporal evolution ofDouglas fir tree stems tending strongly towards cylinders at early ages andtending away with increased competition.

Inference on 3D Procrustes means: tree bole growth, rank-deficient diffusion tensors and perturbation models