Let Xᵢ ≥ 0 be independent, i = 1, ⋯, n, and X^ₙ = max(X₁, ⋯, Xₙ). Let $t(c) (s(c))$ be the threshold stopping rule for X₁, ⋯, Xₙ, defined by t(c) = smallest i for which Xᵢ ≥ c(s(c) = smallest i for which Xᵢ > c), = n otherwise. Let m be a median of the distribution of X^ₙ. It is shown that for every n and X either EX^ₙ ≤ 2EXₜ₍ₘ₎ or EX^ₙ ≤ 2EXₛ₍ₘ₎. This improves previously known results, [1], [4]. Some results for i.i.d. Xᵢ are also included.
No takes yet. Share an insight, caveat, or question.
Ester Samuel‐Cahn (1984) studied this question.