A motivation for $1/ \sqrt{n}$-shrinking neighborhoods

Abstract

In this paper we give a motivation for the shrinking rate $1/ \sqrt{n}:$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 $1/ \sqrt{n}$. 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 $1/ \sqrt[4]{n}$, but leads to the same optimality theory as in the corresponding $1/ \sqrt{n}$ setup.