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Abstract:
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While there are now several procedures available for obtaining the significance of variables or groups of variables in high-dimensional linear models, we do not have a corresponding array of diagnostic tests to check whether the high-dimensional linear model is itself correct. In this talk, I will introduce a family of goodness of fit tests, which we call Residual Prediction (RP) tests, that aim to fill this gap in the practitioner's toolbox. One of our contributions is a method which under certain assumptions can simulate from (essentially) the exact distribution of the scaled residuals following a Lasso fit, under the null hypothesis that the high-dimensional linear model is correct. This allows a whole range of tests to be constructed that are tailored to detect specific departures from the null model, and critical values can be determined easily by simulation without having to resort to complicated analyses of distributions of test statistics. RP tests can be used to test for significance of groups of variables as a special case, but can also be designed to test for as diverse model misspecifications as heteroscedasticity and different types of nonlinearity.
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