The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
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...The proof of Anderson’s theorem hinges on a property of a function gK,L on Rn associated with convex bodies K and L in Rn, defined by gK,L(x) = V (K ∩ (L + x)) ....
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...Anderson’s theorem has many applications in probability and statistics, where, for example, it can be applied to show that certain statistical tests are unbiased....
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...In 1955, Anderson [2] used the Brunn-Minkowski inequality in his work on multivariate unimodality....
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...Applications to probability and statistics In 1955, Anderson [2] used the Brunn-Minkowski inequality in his work on multivariate unimodality....
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...Such applications are treated in Section 11, along with related consequences of Anderson’s theorem on multivariate unimodality, the proof of which employs the Brunn-Minkowski inequality....
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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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