A SIGNIFICANCE TEST FOR THE LASSO.
A SIGNIFICANCE TEST FOR THE LASSO.
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DOI：
10.1214/13-aos1175
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发表时间：
2014-04
影响因子：
4.5
通讯作者：
Tibshirani R
中科院分区：
文献类型：
--
作者：
Lockhart R;Taylor J;Tibshirani RJ;Tibshirani R
在稀疏线性回归设置中，我们考虑测试进入当前lasso模型的预测变量的显著性，在沿着lasso解路径沿着访问的模型序列中。我们提出了一个简单的基于lasso拟合值的检验统计量，称为协方差检验统计量，并表明当真实模型是线性的时，该统计量在零假设下具有Exp（1）渐近分布（零假设是所有真正有效的变量都包含在当前套索模型中）。我们对第一个进入模型的预测变量的特殊情况（即，针对全局空值检验单个显著预测变量）仅需要对预测矩阵X的弱假设。另一方面，我们对套索路径中的一般步骤的证明对X和生成模型进行了进一步的技术假设，但仍然允许重要的高维情况p > n，并且不一定要求当前套索模型实现真正有效变量的完美恢复。当然，为了测试两个嵌套线性模型之间的附加变量的显著性，通常使用卡方检验，将残差平方和（RSS）的下降与 分布但是，当这个附加变量不是固定的，并且已经被自适应地或绿色地选择时，这个测试就不再合适了：自适应性使得RSS随机数的下降比 在零假设下我们的分析明确说明了自适应性，因为它必须，因为套索建立一个自适应序列的线性模型作为调整参数λ减少。在该分析中，收缩起着关键作用：尽管自适应地选择了额外的变量，但是由于 点球因此，检验统计量（基于套索拟合值）在某种意义上被这两个相反的属性（自适应性和收缩性）所平衡，并且其零分布是易于处理的，并且是渐近的Exp（1）。
In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an Exp(1) asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first predictor to enter the model (i.e., testing for a single significant predictor variable against the global null) requires only weak assumptions on the predictor matrix X. On the other hand, our proof for a general step in the lasso path places further technical assumptions on X and the generative model, but still allows for the important high-dimensional case p > n, and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables. Of course, for testing the significance of an additional variable between two nested linear models, one typically uses the chi-squared test, comparing the drop in residual sum of squares (RSS) to a distribution. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate: adaptivity makes the drop in RSS stochastically much larger than under the null hypothesis. Our analysis explicitly accounts for adaptivity, as it must, since the lasso builds an adaptive sequence of linear models as the tuning parameter λ decreases. In this analysis, shrinkage plays a key role: though additional variables are chosen adaptively, the coefficients of lasso active variables are shrunken due to the penalty. Therefore, the test statistic (which is based on lasso fitted values) is in a sense balanced by these two opposing properties—adaptivity and shrinkage—and its null distribution is tractable and asymptotically Exp(1).