A local maximal inequality under uniform entropy
نویسندگان
چکیده
منابع مشابه
A local maximal inequality under uniform entropy.
We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant δ. The bound is expressed in the uniform entropy integral of the class at δ. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions.
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Proof. To see L2-convergence, we will prove that (Sn)n>1 is Cauchy in L 2. It is enough to show that ||Sn − Sm||2 < ε for all n,m > N(ε). Suppose n > m, then we obtain ||Sn − Sm||2 = E(Sn − Sm) = E(Xm+1 +Xm+2 + · · ·+Xn) = Var(Xm+1 +Xm+2 + · · ·+Xn) = Var(Xm+1) + Var(Xm+2) + · · ·+ Var(Xn). For any ε > 0, there exists N = N(ε) with ∑∞ i=N Var(Xi) < ε 2, thus we have ||Sn−Sm||2 < ε for all n,m >...
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ژورنال
عنوان ژورنال: Electronic Journal of Statistics
سال: 2011
ISSN: 1935-7524
DOI: 10.1214/11-ejs605