## The Annals of Statistics

- Ann. Statist.
- Volume 15, Number 4 (1987), 1482-1490.

### Comparisons of Optimal Stopping Values and Prophet Inequalities for Negatively Dependent Random Variables

Yosef Rinott and Ester Samuel-Cahn

#### Abstract

Let $\mathbf{Y} = (Y_1, \cdots, Y_n)$ be random variables satisfying the weak negative dependence condition: $P(Y_i < a_i\mid Y_1 < a_1, \cdots, Y_{i-1}) \leq P(Y_i < a_i)$ for $i = 2, \cdots, n$ and all constants $a_1, \cdots, a_n$. Let $\mathbf{X} = (X_1, \cdots, X_n)$ have independent components, where $X_i$ and $Y_i$ have the same marginal distribution, $i = 1, \cdots, n$. It is shown that $V(\mathbf{X}) \leq V(\mathbf{Y})$, where $V(\mathbf{Y}) = \sup \{EY_t: t \text{is a stopping rule for} Y_1,\cdots, Y_n\}$. Also, the classical inequality which for nonnegative variables compares the expected return of a prophet $E\{Y_1 \vee \cdots \vee Y_n\}$ with that of the statistician $V(\mathbf{Y})$, i.e., $E\{Y_1 \vee \cdots \vee Y_n\} < 2V(\mathbf{Y})$, holds for nonnegative $\mathbf{Y}$ satisfying the negative dependence condition. Moreover, this inequality can be obtained by an explicitly described threshold rule $t(b)$, i.e., $E\{Y_1 \vee \cdots \vee Y_n\} < 2EY_{t(b)}$. Generalizations of this prophet inequality are given. Extensions of the results to infinite sequences are obtained.

#### Article information

**Source**

Ann. Statist., Volume 15, Number 4 (1987), 1482-1490.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350605

**Digital Object Identifier**

doi:10.1214/aos/1176350605

**Mathematical Reviews number (MathSciNet)**

MR913569

**Zentralblatt MATH identifier**

0639.60052

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60E15: Inequalities; stochastic orderings

**Keywords**

Prophet inequality optimal stopping negative dependence negative association sampling without replacement

#### Citation

Rinott, Yosef; Samuel-Cahn, Ester. Comparisons of Optimal Stopping Values and Prophet Inequalities for Negatively Dependent Random Variables. Ann. Statist. 15 (1987), no. 4, 1482--1490. doi:10.1214/aos/1176350605. https://projecteuclid.org/euclid.aos/1176350605