Conservative confidence intervals on multiple correlation coefficient for high-dimensional elliptical data using random projection methodology

So called multiple correlation coefficient (MCC) is a measure of linear relationship between a given variable and set of covariates. In the multiple correlation and regression analysis, it is common practice to construct a confidence interval for the population MCC. In high-dimensional data settings, by which the data dimension p is much larger than the sample size n, due to the singularity of the sample covariance matrix, the classical confidence intervals for the MCC are no longer useable. For high-dimensional elliptical data, some (conservative) confidence intervals for the population MCC are presented using the random projection methodology. To evaluate and compare the performance of the proposed confidence intervals, some simulations are conducted in terms of the coverage probability and average interval length. Experimental validation of the proposed intervals is carried out on two real gene expression datasets.