Generalized Likelihood Ratio Statistics and Wilks Phenomenon
广义似然比统计量与 Wilks 现象AI 解读 ↓
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广义似然比统计量与 Wilks 现象
- 研究问题
- 在非参数函数估计中，如何克服最大似然比统计量不存在、难以计算或不具最优性的问题，并构造有效检验？
- 方法与思路
- 作者提出基于适当非参数估计量的广义似然统计量，在多种模型和原假设下研究极限分布，并考虑自适应平滑参数。
- 材料陈述的贡献
- 摘要称，一类统计量具有渐近无分布性及卡方极限，达到所述最优检测速率，并可通过平滑参数选择实现自适应最优性。
- 阅读关注点
- 应核对适用的原假设与模型、非参数估计量和平滑参数条件，以及卡方极限、最优速率和自适应结论的具体定义。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；具体正则条件、校准方式及有限样本表现未说明，不能推断刊本变化或普遍最优性。
原文摘要
Likelihood ratio theory has had tremendous success in parametric inference, due to the fundamental theory of Wilks. Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to find and can not be optimal as shown in this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. A new Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow χ2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and generalized varying coefficient models. We further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster. They can even be adaptively optimal in the sense of Spokoiny by using a simple choice of adaptive smoothing parameter. Our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation.