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The bracketing Hellinger e—entropy H(e, F) is defined as:

H(e, F) = ln N(e, F). We assume that for a constant c e

Theorem 1 (W.H.Wong and X.Shen). Suppose that f (x) is the true density. Then under condition (10), there exist positive constants C1,C2,C such that

Fi sup TT IT^T ^ expi-cmE2} \ < Cexp{-C2nE2}. (11)

Now suppose that qn = qn(y; x) is a nonnegative function qn : Yn x Y — R1, such that j qn(y; x) j(dx) = 1 (jn-a.s. on y).

Here y = (yi..., yn) € Yn, x € Y, ¡n = j x ... x ¡. We assume that

So, the random function fn(x) = qn(Yi,.. .,Yn; x) may be considered as an estimator for f.

Suppose that f (x) is the true density, and that function qn(y; ■) satisfies the condition

17 qn(y; yj) > exp {-cinE2} (¡n-a.s. on y). (13)

Lemma 1. Suppose that f (x) is the true density, fn(x) is an estimator for f with values in F. Then under conditions (11),(13) for some positive constants c,C

Proof. Let fn( ■) = qn(Yl,...,Yn', ■). We may assume that qn(y;■) G Y. Denote by d(y) the Hellinger distance between qn(y; ■ ) and f (■ ),

It is clear that

{y : d{y) > e, } C { y : sup TT > exp {-cme2} \ .

Therefore,

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