Generalized likelihood ratio statistics and Wilks phenomenon
Generalized likelihood ratio statistics and Wilks phenomenon
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DOI：
10.1214/aos/996986505
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发表时间：
2001-02-01
影响因子：
4.5
通讯作者：
Zhang, J
中科院分区：
文献类型：
--
作者：
Fan, JQ;Zhang, CM;Zhang, J
由于威尔克斯的基本理论，似然比理论在参数推断中取得了巨大的成功。然而，目前还没有一个通用的方法来进行基于函数估计的非参数推断。在非参数函数估计中，极大似然比检验统计量一般不存在。即使它们存在，它们也很难找到，并且不能像本文所示的那样是最优的。为了克服非参数极大似然比统计量的不足，我们引入了广义似然统计量。一个新的S Wilks现象被揭开。本文证明了一类基于适当的非参数估计的广义似然统计量在零假设下是渐近分布自由的，并且对于许多有用的假设和各种有用的模型，包括高斯白色噪声模型、非参数回归模型、变系数模型和广义变系数模型，都服从χ（2）分布.我们进一步证明了广义似然比统计量是渐近最优的，在这个意义上，他们实现了最佳的收敛速度由Ingster。它们甚至可以通过使用自适应平滑参数的简单选择在Spokoiny意义上自适应地最优。我们的工作表明，广义似然比统计量确实是通用的和强大的基于函数估计的非参数检验问题。
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 S 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 chi (2)-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.