A motivation for 1/√n- shrinking neighborhoods

In this paper we give a motivation for the shrinking rate 1/root n: let p (0) and q (n) be the outlier probability under the ideal model, and some member of a neighborhood about this ideal model of radius r (n) , respectively. Assuming n i.i.d. observations, the critical rate of r (n) may be defined such that the minimax test for outlier probability q (n) = p(0) versus q (n) > p(0) has asymptotic error probabilities bounded away from 0 and 1/2. Summarizing the neighborhoods to their upper probability, this leads to r (n) of the exact rate 1/root n. The result makes precise and simplifies ideas in Bickel (1981), Rieder (1994), and Huber (1997). Considering general probabilities of exact Hellinger distance r (n) to P, this shrinking rate translates into 1/(4)root n, but leads to the same optimality theory as in the corresponding 1/root n setup.