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Factor analysis (FA) and principal component analysis (PCA) are popularstatistical methods for summarizing and explaining the variability inmultivariate datasets. By default, FA and PCA assume the number of componentsor factors to be known \emph{a priori}. However, in practice the users firstestimate the number of factors or components and then perform FA and PCAanalyses using the point estimate. Therefore, in practice the users ignore anyuncertainty in the point estimate of the number of factors or components. Fordatasets where the uncertainty in the point estimate is not ignorable, it isprudent to perform FA and PCA analyses for the range of positive integer valuesin the confidence intervals for the number of factors or components. We addressthis problem by proposing a subsampling-based data-intensive approach forestimating confidence intervals for the number of components in FA and PCA. Westudy the coverage probability of the proposed confidence intervals and providenon-asymptotic theoretical guarantees concerning the accuracy of the confidenceintervals. As a byproduct, we derive the first-order \emph{Edgeworth expansion}for spiked eigenvalues of the sample covariance matrix when the data matrix isgenerated under a factor model. We also demonstrate the usefulness of ourapproach through numerical simulations and by applying our approach forestimating confidence intervals for the number of factors of the genotypingdataset of the Human Genome Diversity Project.

Confidence Intervals for the Number of Components in Factor Analysis and Principal Components Analysis via Subsampling