ARE DISCOVERIES SPURIOUS? DISTRIBUTIONS OF MAXIMUM SPURIOUSCORRELATIONS AND THEIR APPLICATIONS

摘要

Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries by such data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given certain number of predictors, namely, the distribution of the correlation of a response variable >Y with the best s linear combinations of p covariates >X, even when >X and >Y are independent. When the covariance matrix of >X possesses the restricted eigenvalue property, we derive such distributions for both finite s and diverging s, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of >X. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such asimple bootstrap approach. The results are further extended to the situationwhere residuals are from regularized fits. Our approach is then applied toconstruct the upper confidence limit for the maximum spurious correlation andtesting exogeneity of covariates. The former provides a baseline for guardingagainst false discoveries due to data mining and the latter tests whether ourfundamental assumptions for high-dimensional model selection are statisticallyvalid. Our techniques and results are illustrated by both numerical examples andreal data analysis.