Comparison of Threshold Stop Rules and Maximum for Independent Nonnegative Random Variables

Abstract

Let $X_i\ge 0$ be independent, $i=1,\cdots ,n$, and $X_n^{\ast }=max(X_1,\cdots ,X_n)$. Let $t(c)(s(c))$ be the threshold stopping rule for $X_1,\cdots ,X_n$, defined by $t(c)=\text{smallest}i$ for which $X_i\ge c(s(c)=\text{smallest}i$ for which $X_i>c),=n$ otherwise. Let $m$ be a median of the distribution of $X_n^{\ast }$. It is shown that for every $n$ and $\underset{―}{X}$ either $E{X}_n^{\ast }\le 2E{X}_{t(m)}$ or $E{X}_n^{\ast }\le 2E{X}_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.