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In this paper we give a motivation for the shrinking rate
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
. 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
, but leads to the same optimality theory as in the corresponding
setup. 相似文献
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