在变量选择中，应用范围最广泛的准则莫过于AIC、BIC和Cp准则，而且有理论证明BIC在经典回归模型中是强相合的。随着社会的进步以及计算机技术的发展，高维数据越来越普遍，对于高维数据，AIC、BIC和Cp准则是否依然适用便成为了摆在研究者面前一个亟待解决的问题。本项目将主要研究在大维架构下，即样本维数与样本量同阶趋于无穷时，AIC、BIC和Cp准则的强相合性。其中包括如下三个方面：（1）高维线性回归模型中AIC、BIC与Cp准则的强相合性；（2）高维主成分分析中AIC、BIC与Cp准则的强相合性；（3）高维典型相关分析中AIC、BIC与Cp准则的强相合性。特别地，我们将证明，AIC和Cp准则在满足某些高维条件时是强相合的，而BIC是不相合的，这与经典结论完全相悖。同时，在此基础上，我们也将对于其他的衍生准则进行研究，例如CAIC、MAIC、HAIC、MCp等。

In statistical variable selection, the most widely used criterions should be AIC, BIC and Cp criterion. And it has been proved that BIC is strong consistent in traditional regression models. With social progress and the development of computer technology, high dimensional data are more and more common. Thus, for the high dimensional data, whether AIC, BIC and Cp criterion are also applicable is the most pressing problem. Therefore, in this project, we will investigate the strong consistency of AIC, BIC and Cp criterion under the large dimensional settings, which assume that the rates of dimension and the sample size tending to infinity have the same order. We will show the following three kinds of results: (1) the strong consistency of AIC, BIC and Cp criterion for high dimensional regression model; (2) the strong consistency of AIC, BIC and Cp criterion for high dimensional principle component analysis; (3) the strong consistency of AIC, BIC and Cp criterion for high dimensional canonical correlation analysis. More specially, we will show that under some high dimensional conditions, AIC and Cp criterion are strong consistent, but BIC is not, which are converse to the traditional results. In addition, we will also consider other criterions which are developed from AIC, BIC and Cp criterion, such as CAIC、MAIC、HAIC、MCp and so on.