Inference on multiple correlation coefficients with moderately high dimensional data

When the multiple correlation coefficient is used to measure how strongly a given variable can be linearly associated with a set of covariates, it suffers from an upward bias that cannot be ignored in the presence of a moderately high dimensional covariate. Under an independent component model, we derive an asymptotic approximation to the distribution of the squared multiple correlation coefficient that depends on a simple correction factor. We show that this approximation enables us to construct reliable confidence intervals on the population coefficient even when the ratio of the dimension to the sample size is close to unity and the variables are non-Gaussian.