Mode testing in difficult cases
困难情形下的众数检验AI 解读 ↓
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困难情形下的众数检验
- 研究问题
- 检验单峰对双峰时，若单峰密度还包含一个平坦肩部，通常的校准是否失效，如何获得渐近正确的检验水平？
- 方法与思路
- 作者从恰有一个众数和一个肩部的模板密度重抽样，校准带宽检验和 dip／超额质量检验，并利用统计量比值的渐近性质。
- 材料陈述的贡献
- 摘要称，按含肩部情形校准的检验用于无肩部情形通常偏保守，反向使用则倾向过度拒绝；所提校准还具有良好适应性。
- 阅读关注点
- 应核对肩部的数学定义、模板密度如何确定、渐近因子化成立的条件，以及不同峰形下的检验水平与功效。
- 阅读边界
- 依据版本未注明的 OpenAlex 索引摘要；模板构造、正则条件和适应性度量未说明，不能据此判断任意密度下的有效性。
原文摘要
Usually, when testing the null hypothesis that a distribution has one mode against the alternative that it has two, the null hypothesis is interpreted as entailing that the density of the sampling distribution has a unique point of zero slope, which is a local maximum. In this paper we argue that a more appropriate null hypothesis is that the density has two points of zero slope, of which one is a local maximum and the other is a shoulder. We show that when a test for a mode-with-shoulder is properly calibrated, so that it has asymptotically correct level, it is generally conservative when applied to the case of a mode without a shoulder. We suggest methods for calibrating both the bandwidth and dip-excess mass tests in the setting of a mode with a shoulder. We also provide evidence in support of the converse: a test calibrated for a single mode without a shoulder tends to be anticonservative when applied to a mode with a shoulder. The calibration method involves resampling from a ‘‘template’’ density with exactly one mode and one shoulder. It exploits the following asymptotic factorization property for both the sample and resample forms of the test statistic: all dependence of these quantities on the sampling distribution cancels asymptotically from their ratio. In contrast to other approaches, the method has very good adaptivity properties.