Complete Classes for Sequential Tests of Hypotheses
序贯假设检验的完全类AI 解读 ↓
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序贯假设检验的完全类
- 研究问题
- 当序贯检验的损失同时包含终端决策错误和观测成本时，如何刻画完全类或本质完全类，并处理假设集合闭包的作用？
- 方法与思路
- 作者先研究原假设和备择假设闭包之间检验的完全类，再利用风险连续性联系原问题，并分别处理三类模型与成本设置。
- 材料陈述的贡献
- 摘要称，闭包问题的完全类可导出原问题的本质完全类，风险连续时可得到完全类；首类模型中贝叶斯检验构成完全类。
- 阅读关注点
- 应核对完全类与本质完全类的定义、风险连续性条件，以及紧致性、首个观测成本和指数族结构在三类结果中的区别。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；具体技术条件未完整给出，不能将首类模型的贝叶斯完全类结论推广到所有序贯检验。
原文摘要
We consider problems of sequential testing when the loss function is the sum of a component due to an error in the terminal decision and a cost of observation component. In all cases we establish a characterization of a complete class or an essentially complete class. In order to obtain such results for testing a null hypothesis against an alternative hypothesis we establish complete class results for testing the closure of the null hypothesis against the closure of the alternative hypothesis. A complete class for testing closure of null against closure of alternative is an essentially complete class for testing null against alternative. Furthermore, a complete class for testing closure of null against closure of alternative is a complete class for testing null against alternative when the risks have certain continuity properties. Such continuity properties do hold in many cases. Three models are treated. The first is when the closure of the null space is compact and the cost of the first observation is positive. Under very unrestrictive conditions it is shown that the Bayes tests form a complete class. This result differs considerably from most fixed sample analogues that have been studied. The second model is when the closure of the null space is compact, the distributions are exponential family, and the cost of the first observation is zero. The third model is for the one dimensional exponential family case when the hypotheses are one sided.