Non-parametric lower bounds and information functions

We argue that common features of non-parametric estimation appear in parametric cases as well if there is a deviation from the classical regularity condition. Namely, in many non-parametric estimation problems (as well as some parametric cases) unbiased finite-variance estimators do not exist; neither estimator converges locally uniformly with the optimal rate; there are no asymptotically unbiased with the optimal rate estimators; etc.. We argue that these features naturally arise in particular parametric subfamilies of non-parametric classes of distributions. We generalize the notion of regularity of a family of distributions and present a general regularity condition, which leads to the notions of the information index and the information function. We argue that the typical structure of a continuity modulus explains why unbiased finite-variance estimators cannot exist if the information index is larger than two, while in typical non-parametric situations neither estimator converges locally uniformly with the optimal rate. We present a new result on impossibility of locally uniform convergence with the optimal rate