The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
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Cites methods from "The integral of a symmetric unimoda..."
...It follows from Anderson (1955) that P { ∫ 1 0 ‖Ḡ0(x)‖2dx ≥ t } ≥ P { ∫ 1 0 ‖Ḡ(x)‖2dx ≥ t } for all t, and therefore the level of rejection is at least α when aN = N−1/2....
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...We thank Professor Davar Khoshnevisan for directing us to the paper of Anderson (1955). This research has been partially supported by NSF Grants DMS–1462067 and DMS–1305858....
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...We thank Professor Davar Khoshnevisan for directing us to the paper of Anderson (1955)....
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18 citations
18 citations
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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