Abstract We consider a problem of sequential testing a simple hypothesis against a simple alternative, based on observations of a discrete-time stochastic process in the presence of a random horizon H. At any time n of the experiment, the statistician is only informed whether H > n or not. In this latter case, the experiment should be terminated and the final decision on the acceptance or rejection of the hypothesis should be taken on the basis of the available observations ( ). H is assumed to be independent of the observations, and its distribution is known to the statistician. Under the random horizon, we consider a variant of the modified Kiefer-Weiss problem: given restrictions on the probabilities of errors, minimize the average sample size calculated under the assumption that the observations follow a fixed distribution, not necessarily one of those hypothesized. Under suitable conditions on the process and/or the horizon, we characterize the structure of all optimal sequential tests in this problem. Then, we apply these results to characterize optimal tests in the case of independent observations. On the basis of the general theory, more specific results are obtained for independent and identically distributed (i.i.d.) observations with a geometrically distributed horizon. In a simple sampling model, we solve the Kiefer-Weiss problem under the random horizon model. We also discuss the questions of Wald-Wolfowitz optimality in the presence of the random horizon. In particular, we show that the stopping rules of the optimal tests, minimizing the average sample size under one of the hypotheses, are randomized versions of those of Wald’s sequential probability ratio tests.

中文翻译：

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随机范围下的顺序假设检验

摘要 我们考虑了一个问题，根据对存在随机范围 H 的离散时间随机过程的观察，针对一个简单的替代方案顺序检验一个简单的假设。在实验的任何时间 n，统计学家只被告知 H > n 与否。在后一种情况下，应终止实验，并根据可用的观察结果（ ）做出接受或拒绝假设的最终决定。假设 H 独立于观测值，并且统计学家知道其分布。在随机范围下，我们考虑修改后的 Kiefer-Weiss 问题的变体：给定错误概率的限制，在假设观测值遵循固定分布的情况下最小化平均样本量，不一定是那些假设之一。在过程和/或范围的合适条件下，我们表征了该问题中所有最优顺序测试的结构。然后，我们将这些结果应用于在独立观察的情况下表征最佳测试。在一般理论的基础上，对于具有几何分布的视界的独立同分布（iid）观测，得到了更具体的结果。 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 在一个简单的抽样模型中，我们解决了随机水平模型下的 Kiefer-Weiss 问题。我们还讨论了存在随机视界的 Wald-Wolfowitz 最优性问题。特别是，我们展示了最优测试的停止规则，在假设之一下最小化平均样本量，