text: Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the conditional distribution of the singular values of a Wishart matrix and derive exact hypothesis tests and confidence intervals for the true signals. Our paper is based on the approach of Taylor, Loftus and Tibshirani (2013) for testing the global null: we generalize it to test for any number of principal components, and derive an integrated version with greater power. In simulation studies we find that our proposed methods compare well to existing approaches.
  @cite_0: We consider a multiple hypothesis testing setting where the hypotheses are ordered and one is only permitted to reject an initial contiguous block, H_1, ,H_k, of hypotheses. A rejection rule in this setting amounts to a procedure for choosing the stopping point k. This setting is inspired by the sequential nature of many model selection problems, where choosing a stopping point or a model is equivalent to rejecting all hypotheses up to that point and none thereafter. We propose two new testing procedures, and prove that they control the false discovery rate in the ordered testing setting. We also show how the methods can be applied to model selection using recent results on p-values in sequential model selection settings.
  @cite_3: Abstract An expression is derived for the null density of the Bartlett test statistic for testing equality of variances of n normal populations, when random samples not necessarily of the same size are taken. The expression permits computation of exact Bartlett critical values. As a generalization, an expression is derived for the density of the normalized ratio of an arbitrarily weighted geometric mean to the unweighted arithmetic mean, based on a sample of n independently distributed gamma random variables having common scale parameter but different shape parameters.
  @cite_7: Tables. Commonly Used Notation. 1. The Multivariate Normal and Related Distributions. 2. Jacobians, Exterior Products, Kronecker Products, and Related Topics. 3. Samples from a Multivariate Normal Distribution, and the Wishart and Multivariate BETA Distributions. 4. Some Results Concerning Decision-Theoretic Estimation of the Parameters of a Multivariate Normal Distribution. 5. Correlation Coefficients. 6. Invariant Tests and Some Applications. 7. Zonal Polynomials and Some Functions of Matrix Argument. 8. Some Standard Tests on Covariance Matrices and Mean Vectors. 9. Principal Components and Related Topics. 10. The Multivariate Linear Model. 11. Testing Independence Between k Sets of Variables and Canonical Correlation Analysis. Appendix: Some Matrix Theory. Bibliography. Index.
summary: Our method is motivated by the Kac-Rice test , an exact method for testing and constructing confidence intervals for signals under the global null hypothesis in adaptive regression. Our work corresponds to the Kac-Rice test under the global null scenario, in a penalized regression minimizing the Frobenius norm with a nuclear norm penalty. In this paper, we extend the test and confidence intervals to the general case with improved power. The resulting statistic uses the survival function of the conditional distribution of the eigenvalues of a Wishart matrix. In the context of inference based on the distribution of eigenvalues, @cite_7 and, more recently, @cite_0 have proposed methods for testing essentially the same hypothesis as in this paper. Both @cite_7 and @cite_0 benefit from an asymptotic distribution of the test statistic: @cite_7 forms a likelihood ratio test with the asymptotic Chi-square distribution. @cite_0 use the Tracy-Widom law, which is the asymptotic distribution of the largest eigenvalue of a Wishart matrix, incorporating the result of @cite_3 . We provide a method for constructing confidence intervals in addition to hypothesis testing, with the procedures being exact.