The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
Citations
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87 citations
Cites background from "The integral of a symmetric unimoda..."
...This follows from a more general inequality which is somewhat reminiscent of Anderson’s lemma [Anderson (1955)]: THEOREM 3.5....
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...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information....
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...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities....
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...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647. Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications....
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...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647. Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econometric Theory 26 445-469. Balabdaoui, F., Rufibach, K. and Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density....
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87 citations
86 citations
Cites background from "The integral of a symmetric unimoda..."
...It was shown by Anderson [1] that the integral of an even unimodal function over translates of a symmetric convex region is maximal if the center of symmetry is moved to the origin....
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...If q ∈ [1, 2), then an even finite Borel measure μ on S1 is a qth dual curvature measure if and only if μ(S1 ∩ L) μ(S1) < 1 q for every one-dimensional subspace L of R2....
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...It was shown by Anderson [1] that the integral of...
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81 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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