Goodness-of-fit tests for high dimensional linear models
Goodness-of-fit tests for high dimensional linear models
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DOI：
10.1111/rssb.12234
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发表时间：
2018-01-01
影响因子：
5.8
通讯作者：
Buhlmann, Peter
中科院分区：
文献类型：
--
作者：
Shah, Rajen D.;Buhlmann, Peter
我们提出了一个在低维线性模型和高维线性模型中构建拟合优度检验的框架。我们主张使用回归方法对数据进行普通最小二乘或套索拟合后的比例残差，并使用一些预测误差的代理作为最终的检验统计量。我们称这一族为残差预测检验。我们证明了模拟可以用来在低维环境下获得这类检验的临界值，并利用理论结果和广泛的数值研究证明了当考虑高维线性模型时，某种形式的参数自举也可以做同样的事情。我们表明，残差预测检验可以作为特例用于检验群体或单个变量的重要性，在这里，它们比最先进的方法更好，但我们也认为，它们可以被设计成测试各种模型错误规范，如异方差和非线性。
We propose a framework for constructing goodness-of-fit tests in both low and high dimensional linear models. We advocate applying regression methods to the scaled residuals following either an ordinary least squares or lasso fit to the data, and using some proxy for prediction error as the final test statistic. We call this family residual prediction tests. We show that simulation can be used to obtain the critical values for such tests in the low dimensional setting and demonstrate using both theoretical results and extensive numerical studies that some form of the parametric bootstrap can do the same when the high dimensional linear model is under consideration. We show that residual prediction tests can be used to test for significance of groups or individual variables as special cases, and here they compare favourably with state of the art methods, but we also argue that they can be designed to test for as diverse model misspecifications as heteroscedasticity and non-linearity.