The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
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Cites background from "The integral of a symmetric unimoda..."
...Then for any fixed r > 0, n‖θ̂r,n − θ0‖ = Op(1), n →∞....
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...→ convergence almost surely p → convergence in probability d → convergence in distribution Θ parameter space Z observations space 1A the indicator function of the set A P probability measure Pn empirical probability measure of observations, Pn(A) = 1 n ∑n i=1 1{Zi ∈ A} E expectation with respect to P Zn = Op(1) if for all 2 > 0 there exist a constant M2 for which P (|Zn| ≥ M2) < 2 Zn = op(1) if Zn p → 0...
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...If we take δ = 1/Rn, then the solution of R 2 n( 1 Rn ) ≤ √n, is Rn ≤ n, and we conclude that n‖θ̂r,n − θ0‖ = Op(1)....
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Cites methods from "The integral of a symmetric unimoda..."
...The Šidák’s inequality relies on a theorem of Anderson (Corollary 2 in [73]), which generalizes to the dimension m the following intuitive result: Let f : R→ R be a density function such that 1....
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References
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"The integral of a symmetric unimoda..." refers background in this paper
...In Theorem 1 the equality in (1) holds for k<l if and only if, for every u, (E+y)r\Ku=Er\Ku-\-y....
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...It will be noticed that we obtain strict inequality in (1) if and only if for at least one u, H(u)>H*(u) (because H(u) is continuous on the left)....
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"The integral of a symmetric unimoda..." refers background in this paper
...f ud[H*(u) - H(u)} = b[H*(b) - H(b)] - a[H*(a) - H(a)} (3) " + f [(H(u) - H*(u)]du....
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...J a Since/(x) has a finite integral over E, bH(b)—>0 as b—>oo and hence also bH*(b)—>0 as b—*<x>; therefore the first term on the right in (3) can be made arbitrarily small in absolute value....
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