Abstract
A sample X1,...,Xn consisting of independent identically distributed vectors in ℝp with zero mean and a covariance matrix Σ is considered. The recovery of spectral projectors of high-dimensional covariance matrices from a sample of observations is a key problem in statistics arising in numerous applications. In their 2015 work, V. Koltchinskii and K. Lounici obtained nonasymptotic bounds for the Frobenius norm \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector Pr were constructed assuming that the moment characteristics of the observations are known. This paper describes a bootstrap procedure for constructing confidence sets for the spectral projector Pr of the covariance matrix Σ from given data. This approach does not use the asymptotical distribution of \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.
Similar content being viewed by others
References
J. Tropp, Found. Comput. Math. 12 (4), 389–434 (2012).
R. Vershynin, “Introduction to the non-asymptotic analysis of random matrices,” in Compressed Sensing: Theory and Applications (Cambridge Univ. Press, Cambridge, 2012), pp. 210–268.
R. van Handel, Trans. Am. Math. Soc. 369 (11), 8161–8178 (2017).
V. Koltchinskii and K. Lounici, “Concentration inequalities and moment bounds for sample covariance operators” (2015). ArXiv:1405.2468.
V. Koltchinskii and K. Lounici, Ann. Stat. 45 (1), 121–157 (2017).
V. Spokoiny, and M. Zhilova, Ann. Stat. 43 (6), 2653–2675 (2015).
V. Chernozhukov, D. Chetverikov, and K. Kato, Ann. Stat. 41 (6), 2786–2819 (2013).
V. Chernozhukov, D. Chetverikov, and K. Kato, Ann. Probab. 45 (4), 2309–2352 (2017).
A. Naumov, V. Spokoiny, and V. Ulyanov, “Bootstrap confidence sets for spectral projectors of sample covariance” (2017). arXiv:1703.00871.
F. Götze, A. Naumov, V. Spokoiny, and V. Ulyanov, “Large ball probabilities, Gaussian comparison and anti-concentration,” Bernoulli 25 (2019). arXiv:1708.08663v2.
A. Naumov, V. Spokoiny, Yu. Tavyrikov and V. Ulyanov, “Nonasymptotic estimates for the closeness of Gaussian measures on balls”, Dokl. Math. 98 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.A. Naumov, V.G. Spokoiny, V.V. Ulyanov, 2018, published in Doklady Akademii Nauk, 2018, Vol. 482, No. 6.
Rights and permissions
About this article
Cite this article
Naumov, A.A., Spokoiny, V.G. & Ulyanov, V.V. Confidence Sets for Spectral Projectors of Covariance Matrices. Dokl. Math. 98, 511–514 (2018). https://doi.org/10.1134/S1064562418060285
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562418060285