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On estimation in the reduced-rank regression with a large number of responses and predictors

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  • Kargin, Vladislav

Abstract

We consider a multivariate linear response regression in which the number of responses and predictors is large and comparable with the number of observations, and the rank of the matrix of regression coefficients is assumed to be small. We study the distribution of singular values for the matrix of regression coefficients and for the matrix of predicted responses. For both matrices, it is found that the limit distribution of the largest singular value is a rescaling of the Tracy–Widom distribution. Based on this result, we suggest algorithms for the model rank selection and compare them with the algorithm suggested by Bunea, She and Wegkamp. Next, we design two consistent estimators for the singular values of the coefficient matrix, compare them, and derive the asymptotic distribution for one of these estimators.

Suggested Citation

  • Kargin, Vladislav, 2015. "On estimation in the reduced-rank regression with a large number of responses and predictors," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 377-394.
    • Handle: RePEc:eee:jmvana:v:140:y:2015:i:c:p:377-394
    DOI: 10.1016/j.jmva.2015.06.004
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    References listed on IDEAS

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    1. Jean Fortier, 1966. "Simultaneous nonlinear prediction," Psychometrika, Springer;The Psychometric Society, vol. 31(4), pages 447-455, December.
    2. Jean Fortier, 1966. "Simultaneous linear prediction," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 369-381, September.
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    4. Ming Yuan & Ali Ekici & Zhaosong Lu & Renato Monteiro, 2007. "Dimension reduction and coefficient estimation in multivariate linear regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(3), pages 329-346, June.
    5. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    6. Izenman, Alan Julian, 1975. "Reduced-rank regression for the multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 5(2), pages 248-264, June.
    7. Benaych-Georges, Florent & Nadakuditi, Raj Rao, 2012. "The singular values and vectors of low rank perturbations of large rectangular random matrices," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 120-135.
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    Cited by:

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    2. Zhao, Li & Xu, Xingzhong, 2017. "Generalized canonical correlation variables improved estimation in high dimensional seemingly unrelated regression models," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 119-126.

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