## The Annals of Probability

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

Ester Samuel-Cahn

#### Abstract

Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. 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 $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.

#### Article information

Source
Ann. Probab., Volume 12, Number 4 (1984), 1213-1216.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993150

Digital Object Identifier
doi:10.1214/aop/1176993150

Mathematical Reviews number (MathSciNet)
MR757778

Zentralblatt MATH identifier
0549.60036

JSTOR