## The Annals of Probability

### Approximations to Optimal Stopping Rules for Exponential Random Variables

#### Abstract

For $X_1, X_2, \cdots$ i.i.d. with finite mean and $Y_n = \max(X_1, \cdots, X_n) - cn, c$ positive, a number of authors have considered the problem of determining an optimal stopping rule for the reward sequence $Y_n$. The optimal stopping rule can be given explicitly in this case; however, in general its use requires complete knowledge of the distribution of the $X_i$. This paper examines the problem of approximating the optimal expected reward when only partial information about the distribution is available. Specifically, if the $X_i$ are known to be exponentially distributed with unknown mean, stopping rules designed to approximate the optimal rule (which can be used only when the mean is known) are proposed. Under certain conditions the difference between the expected reward using the proposed stopping rules and the optimal expected reward vanishes as $c$ approaches zero.

#### Article information

Source
Ann. Probab., Volume 12, Number 3 (1984), 876-881.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993236

Digital Object Identifier
doi:10.1214/aop/1176993236

Mathematical Reviews number (MathSciNet)
MR744242

Zentralblatt MATH identifier
0544.62076

JSTOR

#### Citation

Martinsek, Adam T. Approximations to Optimal Stopping Rules for Exponential Random Variables. Ann. Probab. 12 (1984), no. 3, 876--881. doi:10.1214/aop/1176993236. https://projecteuclid.org/euclid.aop/1176993236