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

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 $c>0$, 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.