## Abstract

We consider, in the setting of p and n large, sample covariance matrices whose population counterparts follow a spiked population model, i.e., with the exception of the first (largest) few, all the population eigenvalues are equal. We study the asymptotic distribution of the partial maximum likelihood ratio statistic and use it to test for the dimension of the population spike subspace. Furthermore, we extend this to the ultra-high-dimensional case, i.e., p>;n. A thorough study of the power of the test gives a correction that allows us to test for the dimension of the population spike subspace even for values of the limit of p/n close to 1, a setting where other approaches have proved to be deficient.

Original language | English (US) |
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Pages (from-to) | 18-38 |

Number of pages | 21 |

Journal | Journal of Multivariate Analysis |

Volume | 159 |

DOIs | |

State | Published - Jul 1 2017 |

### Bibliographical note

Funding Information:We would like to gratefully thank the Editor and an anonymous referee for her/his comments which have greatly improved our manuscript. This work was supported by the SECTEI grant 2010-072-14, by the UNL grants 500-040, 501-499 and 500-062; by the CONICET grant PIP 742 and by the ANPCYT grant PICT 2012-2590.

## Keywords

- High-dimensional statistics
- Principal component analysis
- Random matrix theory
- Sample covariance matrix
- Spiked population model