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Bayesian inference for spectral projectors of covariance matrix

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  • Silin, Igor
  • Spokoiny, Vladimir

Abstract

Let X1; : : : ;Xn be i.i.d. sample in Rp with zero mean and the covariance matrix . The classic principal component analysis esti- mates the projector P J onto the direct sum of some eigenspaces of by its empirical counterpart bPJ . Recent papers [20, 23] investigate the asymptotic distribution of the Frobenius distance between the projectors k bPJ ??P J k2 . The problem arises when one tries to build a condence set for the true projector eectively. We consider the problem from Bayesian perspective and derive an approximation for the posterior distribution of the Frobenius distance between projectors. The derived theorems hold true for non-Gaussian data: the only assumption that we impose is the con- centration of the sample covariance b in a vicinity of . The obtained results are applied to construction of sharp condence sets for the true pro- jector. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in quite challenging regime.

Suggested Citation

  • Silin, Igor & Spokoiny, Vladimir, 2018. "Bayesian inference for spectral projectors of covariance matrix," IRTG 1792 Discussion Papers 2018-027, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    • Handle: RePEc:zbw:irtgdp:2018027
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    1. Johnstone, Iain M. & Lu, Arthur Yu, 2009. "On Consistency and Sparsity for Principal Components Analysis in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 682-693.
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    8. Kalkbrener, Michael & Packham, Natalie, 2018. "Correlation Under Stress In Normal Variance Mixture Models," IRTG 1792 Discussion Papers 2018-035, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
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    13. Packham, Natalie & Kalkbrener, Michael & Overbeck, Ludger, 2018. "Default probabilities and default correlations under stress," IRTG 1792 Discussion Papers 2018-037, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    14. Kuczmaszewska, Anna & Yan, Ji Gao, 2018. "On complete convergence in Marcinkiewicz-Zygmund type SLLN for random variables," IRTG 1792 Discussion Papers 2018-041, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    15. Koziuk, Andzhey & Spokoiny, Vladimir, 2018. "Toolbox: Gaussian comparison on Eucledian balls," IRTG 1792 Discussion Papers 2018-028, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    16. Chen, Haiqiang & Li, Yingxing & Lin, Ming & Zhu, Yanli, 2018. "A Regime Shift Model with Nonparametric Switching Mechanism," IRTG 1792 Discussion Papers 2018-048, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    17. Yatracos, Yannis G., 2018. "Residual'S Influence Index (Rinfin), Bad Leverage And Unmasking In High Dimensional L2-Regression," IRTG 1792 Discussion Papers 2018-060, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    18. Zbonakova, Lenka & Li, Xinjue & Härdle, Wolfgang Karl, 2018. "Penalized Adaptive Forecasting with Large Information Sets and Structural Changes," IRTG 1792 Discussion Papers 2018-039, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".

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    More about this item

    Keywords

    covariance matrix; spectral projector; principal component analysis; Bernstein-von Mises theorem;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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