APPROXIMATION TO OPTIMAL STOPPING RULES FOR WEIBULL RANDOM VARIABLES WITH UNKNOWN SCALE PARAMETER

Abstract

Let $X_1, X_2, \cdots, X_n, \cdots$ be independent, identically distributed Weibull random variables with an unknown scale parameter $\alpha$. 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 scale parameter $\alpha$. In this paper we propose an adaptive stopping rule that does not depend on the unknown scale parameter $\alpha$ 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.