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Dimension estimation in sufficient dimension reduction: A unifying approach

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  • Bura, E.
  • Yang, J.

Abstract

Sufficient Dimension Reduction (SDR) in regression comprises the estimation of the dimension of the smallest (central) dimension reduction subspace and its basis elements. For SDR methods based on a kernel matrix, such as SIR and SAVE, the dimension estimation is equivalent to the estimation of the rank of a random matrix which is the sample based estimate of the kernel. A test for the rank of a random matrix amounts to testing how many of its eigen or singular values are equal to zero. We propose two tests based on the smallest eigen or singular values of the estimated matrix: an asymptotic weighted chi-square test and a Wald-type asymptotic chi-square test. We also provide an asymptotic chi-square test for assessing whether elements of the left singular vectors of the random matrix are zero. These methods together constitute a unified approach for all SDR methods based on a kernel matrix that covers estimation of the central subspace and its dimension, as well as assessment of variable contribution to the lower-dimensional predictor projections with variable selection, a special case. A small power simulation study shows that the proposed and existing tests, specific to each SDR method, perform similarly with respect to power and achievement of the nominal level. Also, the importance of the choice of the number of slices as a tuning parameter is further exhibited.

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  • Bura, E. & Yang, J., 2011. "Dimension estimation in sufficient dimension reduction: A unifying approach," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 130-142, January.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:1:p:130-142
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    2. Orea, Luis & Growitsch, Christian & Jamasb, Tooraj, 2012. "Using Supervised Environmental Composites in Production and Efficiency Analyses: An Application to Norwegian Electricity Networks," EWI Working Papers 2012-18, Energiewirtschaftliches Institut an der Universitaet zu Koeln (EWI).
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    4. Klaus Nordhausen & Anne Ruiz-Gazen, 2022. "On the usage of joint diagonalization in multivariate statistics," Post-Print hal-04296111, HAL.
    5. Alain Guay, 2020. "Identification of Structural Vector Autoregressions Through Higher Unconditional Moments," Working Papers 20-19, Chair in macroeconomics and forecasting, University of Quebec in Montreal's School of Management.
    6. Pircalabelu, Eugen & Artemiou, Andreas, 2021. "Graph informed sliced inverse regression," Computational Statistics & Data Analysis, Elsevier, vol. 164(C).
    7. Charles Lindsey & Simon Sheather & Joseph McKean, 2014. "Using sliced mean variance–covariance inverse regression for classification and dimension reduction," Computational Statistics, Springer, vol. 29(3), pages 769-798, June.
    8. Shih‐Hao Huang & Kerby Shedden & Hsin‐wen Chang, 2023. "Inference for the dimension of a regression relationship using pseudo‐covariates," Biometrics, The International Biometric Society, vol. 79(3), pages 2394-2403, September.
    9. Chen, Canyi & Xu, Wangli & Zhu, Liping, 2022. "Distributed estimation in heterogeneous reduced rank regression: With application to order determination in sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    10. Guay, Alain, 2021. "Identification of structural vector autoregressions through higher unconditional moments," Journal of Econometrics, Elsevier, vol. 225(1), pages 27-46.
    11. Nordhausen, Klaus & Oja, Hannu & Tyler, David E., 2022. "Asymptotic and bootstrap tests for subspace dimension," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    12. Stephen Babos & Andreas Artemiou, 2020. "Sliced inverse median difference regression," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(4), pages 937-954, December.
    13. Artemiou, Andreas & Tian, Lipu, 2015. "Using sliced inverse mean difference for sufficient dimension reduction," Statistics & Probability Letters, Elsevier, vol. 106(C), pages 184-190.
    14. Wei Luo, 2022. "On efficient dimension reduction with respect to the interaction between two response variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 269-294, April.
    15. Alessandro Barbarino & Efstathia Bura, 2017. "A Unified Framework for Dimension Reduction in Forecasting," Finance and Economics Discussion Series 2017-004, Board of Governors of the Federal Reserve System (U.S.).
    16. Nordhausen, Klaus & Ruiz-Gazen, Anne, 2022. "On the usage of joint diagonalization in multivariate statistics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
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