Taiwanese Journal of Mathematics

APPROXIMATION TO OPTIMAL STOPPING RULES FOR GUMBEL RANDOM VARIABLES WITH UNKNOWN LOCATION AND SCALE PARAMETERS

Tzu-Sheng Yeh and Shen-Ming Lee

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Abstract

An optimal stopping rule is a rule that stops the sampling process at a sample size n that maximizes the expected reward. In this paper we will study the approximation to optimal stopping rule for Gumbel random variables, because the Gumbel-type distribution is the most commonly referred to in discussions of extreme values. Let $X_1, X_2,\cdots X_n,\cdots$ be independent, identically distributed Gumbel random variables with unknown location and scale parameters,$\alpha$ and $\beta$. If we define the reward sequence $Y_n = \max \{X_1,X_2,\cdots,X_n\}-cn$ for $c \gt 0$, the optimal stopping rule for $Y_n$ depends on the unknown location and scale parameters $\alpha$ and $\beta$. We propose an adaptive stopping rule that does not depend on the unknown location and scale parameters and show that the difference between the optimal expected reward and the expected reward using the proposed adaptive stopping rule vanishes as $c$ goes to zero. Also, we use simulation in statistics to verify the results.

Article information

Source
Taiwanese J. Math., Volume 10, Number 4 (2006), 1047-1067.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500403892

Digital Object Identifier
doi:10.11650/twjm/1500403892

Mathematical Reviews number (MathSciNet)
MR2229640

Zentralblatt MATH identifier
05238260

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

Keywords
optimal stopping uniform integrability last times Gumbel distribution

Citation

Yeh, Tzu-Sheng; Lee, Shen-Ming. APPROXIMATION TO OPTIMAL STOPPING RULES FOR GUMBEL RANDOM VARIABLES WITH UNKNOWN LOCATION AND SCALE PARAMETERS. Taiwanese J. Math. 10 (2006), no. 4, 1047--1067. doi:10.11650/twjm/1500403892. https://projecteuclid.org/euclid.twjm/1500403892


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