Statistical significance in high-dimensional linear models

By Peter Buhlmann. 2012

244 citations recorded by the source corpus.

We propose a method for constructing p-values for general hypotheses in a high-dimensional linear model. The hypotheses can be local for testing a single regression parameter or they may be more global involving several up to all parameters. Furthermore, when considering many hypotheses, we show how to adjust for multiple testing taking dependence among the p-values into account. Our technique is based on Ridge estimation with an additional correction term due to a substantial projection bias in high dimensions. We prove strong error control for our p-values and provide sufficient conditions for detection: for the former, we do not make any assumption on the size of the true underlying regression coefficients while regarding the latter, our procedure might not be optimal in terms of power. We demonstrate the method in simulated examples and a real data application.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b8db0ca28c530dd0d0671af5d1a1c238a5a03fee

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Citations in the corpus

High-Dimensional Statistics with a View Toward Applications in Biology (2014, methodology): “We point out the interesting feature that these estimators β̂corr−Ridge or β̂corr−Lasso are not sparse and they have a Gaussian limiting distribution with known covariance matrix, except for the unknown error variance σ2ε (Bühlmann, 2013; Zhang and Zhang, 2011; van de Geer et al., 2013).”
Linear Hypothesis Testing in Dense High-Dimensional Linear Models (2016, background): “Early work typically imposes conditions that guarantee consistent variable selection (see Fan and Li (2001); Zou (2006); Zhao and Yu (2006)) or develops methods that lead to conservative inferential guarantees (e.g. Bühlmann et al. (2013)).”
Exact Post-Selection Inference for Sequential Regression Procedures (2014, background): “Buhlmann (2012) constructs p-values for coefficients in high-dimensional regression”
Handbook of Big Data (2016, background): “Thereby, use of a formal causal model highlights that in many practical data applications, a nonparametric statistical model is appropriate.”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “This includes the literature on debiased or desparsified regularized models, for example Buhlmann (2013), Zhang and Zhang (2014), Javanmard and Montanari (2014), Bühlmann and van de Geer (2015), Dezeure et al. (2016), Zhang and Cheng (2017), van de Geer et al. (2014), Nickl and van de Geer (2013).”
Statistical Learning with Sparsity: The Lasso and Generalizations (2015, background): “) Similar to the right-hand plot, Meinshausen and Bühlmann (2010) pro-”
SLOPE—ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION (2014, background): “Also, when λ1 > 0 and λ2 = · · · = λp = 0, the sorted `1 norm reduces to the `∞ norm (again, up to a multiplicative factor). and Montanari (2013a,b); Bühlmann (2013); Lockhart et al. (2014)), with the exploration of quite different viewpoints.”
Uniform asymptotic inference and the bootstrap after model selection (2015, methodology): “Here, we study the large sample properties of this method, without assuming normality.”
A SIGNIFICANCE TEST FOR THE LASSO (2013, methodology): “Wasserman and Roeder (2009) propose a procedure for variable selection and p-value estimation in high-dimensional linear models based on sample splitting, and this idea was extended by Meinshausen, Meier and Bühlmann (2009). Meinshausen and Bühlmann (2010) propose a generic method using resampling c”
Asymptotic normality and optimalities in estimation of large Gaussian graphical models (2013, background): “See Meinshausen and Bühlmann (2006), Yuan (2010), Cai, Liu and Luo (2011), Cai, Liu and Zhou (2012) and Sun and Zhang (2012a). The irrepresentability condition is no longer needed in Cai, Liu and Luo (2011) and Cai, Liu and Zhou (2012) for support recovery, but the thresholding level for support rec”