n-Shrinking Neighborhoods

In this paper we give a motivation for the shrinking rate 1/ √ n: Let p0 and qn be the outlier probability under the ideal model, and some member of a neighborhood about this ideal model of radius rn, respectively. Assuming n i.i.d. observations, the critical rate of rn may be de ned such that the minimax test for outlier probability qn = p0 vs. qn > p0 has asymptotic error probabilities bounded away from 0 and 1/2. Summarizing the neighborhoods to their upper probability, this leads to rn of the exact rate 1/ √ n. The result makes precise and simpli es ideas in Bickel (1981), Rieder (1994), and Huber (1997). Considering general probabilities of exact Hellinger distance rn to P, this shrinking rate translates into 1/ 4 √ n, but leads to the same optimality theory as in the corresponding 1/ √ n setup. Mathematics Subject Classi cation (1991): 62F35