A Cram´ er-Rao inequality for non differentiable models Paul Rochet Abstract We compute a variance lower bound for unbiased estimators in specified statis- tical models. The construction of the bound is related to the original Cram´ er-Rao bound, although it does not require the differentiability of the model. Moreover, we show our efficiency bound to be always greater than the Cram´ er-Rao bound in smooth models, thus providing a sharper result. 1 Introduction Efficiency theory aims to establish an objective criterion to judge if an estimator is the best possible in a given class. The most famous example is without doubt the Cram´ er-Rao inequality, which states in its simpler form that the variance of an unbiased estimator in a parametric model is not smaller than the inverse of the Fisher information. The inequality was originally stated in [RR45] and has been the foundation of a numerous efficiency theories developped in the literature, such as that due to Le Cam and Haj´ ek (see [H´ aj70], [LC60]) that extend the Cram´ er-Rao inequality to larger models with alternative regularity assumptions. We refer to [BKRW98] and [vdV98] for a survey. In this paper, we introduce a variance lower bound for unbiased estimators in a sta- tistical model. The construction of the bound relies on the same idea as the original Cram´ er-Rao bound, although no regularity conditions of any kind are needed. The ad- vantage of our approach is threefold. First, an efficiency bound can be computed without differentiability conditions on the model nor on the parameter to estimate. Second, the bound is adapted to all types of models: parametric, semiparametric or nonparamet- ric. Finally, the efficiency bound is always greater or equal to the Cram´ er-Rao bound (whenever it is well defined) and thus is more informative. The paper is organized as follows. We define our efficiency bound in Section 2 and we compare its performance to the Cram´ er-Rao bound in differentiable parametric mod- els. We discuss the generalization to semiparametric models in Section 2.2 and provide an asymptotic analysis in Section 2.3. The proofs of our results are postponed to the Appendix. 1 arXiv:1204.2763v1 [math.ST] 12 Apr 2012