广义线性模型的平方根Lasso选择性推断

Selective Inference of Generalized Linear Models by the Square Root Lasso

摘要: 已经有很多人对线性模型的相关问题做出了大量的选择性推断工作。但是，其适用范围并不全面。实际上，我们会遇到很多非正态非连续的数据，并且随机误差项是异方差等方面的问题，故本文将一些选择性推断的工作推广到广义线性模型上。我们建议使用数据雕琢方法，对数据样本先拆分再进行推断。另外，很多的模型变量选择的工作，都是基于已知噪声水平的。但是，在很多实际情况下，误差方差是未知的，而且在高维数据中，对它的估计存在一定难度。本文中，我们使用平方根lasso，进行变量选择，证明出对于广义线性模型，调优参数的选择不受数据中噪声波动的影响。因此，平方根lasso比lasso调优参数选择更方便，应用范围更广泛。模拟结果表明，使用数据雕琢得到的参数的置信区间更小。

Abstract: In view of the fact that many people have done a lot of work on the selective inference of linear models, however, its scope of application is not comprehensive. In fact, we will encounter many problems such as non normal and non continuous data, and the random error term is heteroscedasticity. Therefore, this paper extends some selective inference work to generalized linear model. We suggest that we use the data carving to split the sample first and then infer. In addition, many of the work of model variable selection are based on the known noise level. However, in practical cases, the error variance is unknown, and in high-dimensional data, it is difficult to estimate. In this paper, we use the square root lasso to select variables, and prove that for the generalized linear model, the selection of tuning parameters is not affected by the noise fluctuations in the data. So the square root lasso is more convenient and widely used than the lasso, and the application is more extensive. Simulation shows that we get the shorter interval length by data carving.