Goodness‐of‐fit tests for high dimensional linear models

By Rajen Dinesh Shah, Peter Buhlmann. 2015

54 citations recorded by the source corpus.

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.

Subfields: Statistics and Probability

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Citations in the corpus

The hardness of conditional independence testing and the generalised covariance measure (2018, methodology): “In fact, in this case our test statistic is closely related to that involved in the ANT procedure of Ren et al. [41] and the socalled RP test introduced in Shah and Bühlmann [46], which amount to a regularised partial correlation.”
Invariant Causal Prediction for Nonlinear Models (2017, methodology): “Use a variant of a Residual Prediction (RP) test (Shah and B¨uhlmann, 2015) with an appropriate basis expansion where the main idea is to scale the residuals of a nonlinear regression such that the resulting test statistics is not a function of the noise variance.”
Invariant Causal Prediction for Sequential Data (2017, background): “Shah and Bühlmann (2017) give a general methodology for dealing with such tests.”
Computation of Probability Associated with Anderson–Darling Statistic (2018, methodology): “Alongside the well-known goodness-of-fit test, other methods based for example on entropy estimator [17–19], jackknife empirical likelihood [20], on the prediction of residuals [21], or for testing multilevel survival data [22] or multilevel models with binary outcomes [23] have been reported in the”
High-dimensional simultaneous inference with the bootstrap (2016, background): “…for some inferential tasks,McKeague and Qian (2015) analyze the bootstrap for marginal correlation screening for high-dimensional linear models, and Shah and Bühlmann (2015) consider the use a bootstrap scheme for obtaining the exact distribution of scaled residuals in a high-dimensional linear…”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “While this is a larger literature than can be addressed completely here, it includes early work on model selection [35] and model averaging interpretations [31]; the impossibility results of [38] and [16] on random X and model misspecification; methods based on resampling or sample splitting [18, 19”