A Short Note on Superefficiency

In Le Cam's earlier work on superefficiency, it is proved that if an estimate is superefficient at a given paramter value $\theta$$\_$0/, then there must exist an infinite sequence {$\theta$$\_$n/}) of values(conversing to $\theta$$\_$0/) at which this estimate is worse than M. L. E. for certain classes of loss functions. For one-dimensional cases, these classes of lass functions include squared error loss. However. for multi-dimensional cases, they do not. This note is to give an example where a superefficiest estimator of a multi-dimensional parameter is not inferior to M. L. E. along any sequence ($\theta$$\_$n/) converging to the point of superefficiency with respect to the squared error loss.