On Optimal Stopping

Abstract

This paper considers the following problem. One takes independent and identically distributed observations from a population obeying a probability law $F_\theta(x)$. However, one does not know $F_\theta(x)$. What is known is a family of distribution functions $\Theta = \{F_\theta(x)\}$ and one assumes that there exists a prior probability measure $\mu(d\theta)$ on $\Theta$. At each stage $n = 1, 2, \cdots$ of the sampling one may stop. If one does stop he gets a payoff: $m_n - nC = Y_n$, where $m_n = \text{maximum} (X_1, \cdots, X_n)$ and $C > 0$. One is interested in a procedure which is optimal. In Section 2 we define "procedure" and "optimal procedure." We show that under some conditions an "optimal procedure" exists. The optimal procedure turns out to be: Stop at stage $j$ if $Y_j = \alpha(j)$ where $\alpha(j)$ is a function of $X_1, X_2, \cdots, X_j\mu(d\theta)$. The function $\alpha(j)$, although easy to describe, is quite difficult to calculate, hence we give in Section 3 two other functions which again are functions of $X_1, \cdots, X_j, \mu(d\theta)$ and are somewhat easier to calculate. These functions, denoted by $\beta_1(j), \beta_2(j)$ have the following properties: $\beta_1(j) \leqq \alpha(j) \leqq \beta_2(j)$ and $\beta_2(j) - \beta_1(j) \rightarrow 0$ almost surely as $j \rightarrow \infty$. Furthermore, we give an example for which $Y_j < \alpha(j) \Leftrightarrow Y_j < \beta_1(j) \Leftrightarrow Y_j < \beta_2(j)$. The case for which $\mu(d\theta)$ is degenerate, namely the case for which $F_\theta(x)$ is known to the sampler was solved in [2]. The work done in [1], [2], [3] helped us to obtain the results in Sections 2 and 3. The work done in [4] and [5] deals with similar problems but the approach is different. All relations in this work are understood to hold almost surely unless otherwise specified.