在高维回归模型的统计推断中，一个突出的问题是冗余参数的影响。在低维冗余参数下，似然比检验是一种非常经典有效的推断方法，它有很多优良的性质，其中最重要的似然比检验的渐近分布常常和冗余参数无关，人们称这一性质为Wilks现象。但当冗余参数维数是高维时，似然比检验是否还具有Wilks现象受到怀疑，并且它的功效会受到严重影响，甚至失去检验能力。关于似然比检验在高维情形下的研究，大多局限在广义线性模型下，很少有针对半参数模型的研究。为此，我们提出在构造（广义）似然比检验之前先降低冗余参数(或系数函数)空间的维数的两步检验方法，即DR-GLR检验。本课题的研究内容如下：（1）DR-GLR统计量在高维参数模型下的构造及性质；（2）DR-GLR统计量在高维半参数模型下的构造及性质；（3）DR-GLR检验在非似然情形下的拓展；（4）DR-GLR检验在高维数据下的数值实现和应用。

In high-dimensional regression, a large number of nuisance parameters greatly affect inference for parameters of interest. When the dimension of nuisance parameters in a model is low, the likelihood ratio test (LRT) is an efficient and powerful tool of statistical inference. The important property of the LRT is that its asymptotic null distribution is independent of the nuisance parameters, referred to as the Wilks theorem (or phenomena). However, it is unknown whether the Wilks theorem still holds when the dimension of nuisance parameters is large, especially when the dimension is large than sample size. Even if it holds, the power of LRT is adversely affected by the number of nuisance parameters so that the LRT becomes too low to detect the difference between the null and alternatives. For high-dimensional data, most studies on LRT are limited to linear or generalized linear models. Very few works have focused on LRT for high-dimensional semiparametric models. For this purpose, we propose to first reduce the dimension of nuisance parameters and then construct generalized LRTs. The resulting test is coined as dimension-reduced generalized likelihood ratio (DR-GLR) test. In this proposal, we aim to investigate the following problems: (1) To construct DR-GLR statistics and investigate their theoretical properties in high-dimensional parametric models. (2) To construct DR-GLR statistics in high-dimensional semiparametric models and examine their asymptotic properties. (3) To extend the idea of DR-GLR test to cases where the likelihood function does not exist. (4) To investigate the numerical realization of DR-GLR tests and apply them to a variety of scientific fields.