On the Erratic Behavior of Estimators of N in the Binomial N, p Distribution
二项分布试验次数N估计量的不稳定行为AI 解读 ↓
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二项分布试验次数N估计量的不稳定行为
- 研究问题
- 为何二项分布试验次数的极大似然与矩估计会对微小数据变动极其敏感，而通常渐近正态理论难以揭示这种行为？
- 方法与思路
- 作者发展替代渐近理论，在与若干难处理数据集相符的参数配置下，比较经典估计量与Carroll–Lombard估计量的极限分布。
- 材料陈述的贡献
- 摘要称这些配置下极大似然与矩估计量的极限分布可具有柯西式重尾，而替代估计量具有轻尾，从而解释两类方法的行为差异。
- 阅读关注点
- 应核对替代渐近框架中参数和样本量如何变化、重尾极限成立的配置，以及极限分布能否解释实际样本中的估计不稳定。
- 阅读边界
- 仅据OpenAlex索引摘要，稿件版本未注明，不能推断刊本变化；重尾与轻尾结论限于摘要所指配置，具体参数条件未说明。
原文摘要
The problem of estimating the number of trials in a Binomial distribution is known to be particularly difficult in certain circumstances. Both the maximum likelihood and method-of-moments estimators can produce very erratic results, being extremely sensitive to minor fluctuations in the data. But existing asymptotic theory for these estimators does not provide any insight into this peculiar behavior, as the theory typically produces asymptotic Normal distributions with respectable means and variances. Likewise, the limiting distributions of other estimators do not convey much information about why those estimators might perform better than their more classical counterparts. In this article we develop an alternative asymptotic theory that clearly and vividly depicts not only the erratic performance of the maximum likelihood and method-of-moments estimators, but also the more regular, robust properties of alternative estimators proposed by Carroll and Lombard. We note that the parameter configurations under which these theoretical results are valid are just those exhibited by well-known, “troublesome” data sets. In these settings, the maximum likelihood and method-of-moments estimators have limiting distributions with very heavy tails, such as the Cauchy, whereas the Carroll-Lombard estimators have limiting distributions with very light tails.