Abstract
Let X1,…,X7 be i.i.d. random variables with a common continuous distribution F, Two parameters, μ(F) = P(X1 < X5 and X1+X4 < X2+X3) and λ(F) = P(X1+X4 < X2+X3 and X1+X7 < X5+X6), which appear in the moments of some rank statistics have been studied by several authors. It is shown that the existing lower bound, 3/10 ≤ μ(F) can be improved to 3/10 < μ(F) and that no further improvement is possible. It is also shown that the existing upper bounds μ(F) ≤ (21/2+6)/24 ≈ 0.30893 and λ(F) ≤ 7/24 ≈ 0.29167 can be improved to [14+(2/3)1/2]/48 ≈ 0.30868 and {7 − [1 − (2/3)1/2]2/4}/24 ≈ 0.29132.
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