Parametric and semiparametric inference for shape: the role of the scale functional

We are considering the problem of efficient inference on the shape matrix of an elliptic distribution with unspecified location and either (a) fully specified radial density, (b) radial density specified up to a scale parameter, or (c) completely unspecified radial density. Bickel in [1] has shown that efficiencies under (b) and (c), while being strictly less than under (a), coincide: the efficiency loss caused by an unspecified radial density thus is entirely due to the non-specification of its scale (scale here is not necessarily measured by standard error, as second-order moments may be infinite). Defining scale however requires the choice of a particular scale functional, a choice which has an impact on efficiency bounds. We provide a closed form expression for this efficiency loss, both in hypothesis testing and in point estimation, as a function of the standardized radial density and the scale functional. We show that this loss is maximum at arbitrarily light tails whereas, under arbitrarily heavy tails, it is arbitrarily close to zero: hence, under very heavy tails, ignoring the scale (ignoring the exact density) asymptotically does not harm inference on shape. However, the same loss is nil, irrespective of the standardized radial density, when the scale functional (in dimension k) is the k-th root of the scatter determinant.