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Nonparametric estimation of large covariance matrices of longitudinal data

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  • Wei Biao Wu
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    Abstract

    Estimation of an unstructured covariance matrix is difficult because of its positive-definiteness constraint. This obstacle is removed by regressing each variable on its predecessors, so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying-coefficient and varying-order regression models. Our framework is similar to the use of increasing-order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series. As an illustration, we adopt Fan & Zhang's (2000) two-step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite. For parsimony a suitable order for the sequence of (auto)regression models is found using penalised likelihood criteria like AIC and BIC. Some asymptotic results for the local polynomial estimators of components of a covariance matrix are established. Two longitudinal datasets are analysed to illustrate the methodology. A simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices. Copyright Biometrika Trust 2003, Oxford University Press.

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    Bibliographic Info

    Article provided by Biometrika Trust in its journal Biometrika.

    Volume (Year): 90 (2003)
    Issue (Month): 4 (December)
    Pages: 831-844

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    Handle: RePEc:oup:biomet:v:90:y:2003:i:4:p:831-844

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    Cited by:
    1. Feng, Yuanhua & Yu, Keming, 2006. "Nonparametric estimation of time-varying covariance matrix in a slowly changing vector random walk model," MPRA Paper 1597, University Library of Munich, Germany.
    2. Xi Luo, 2011. "Recovering Model Structures from Large Low Rank and Sparse Covariance Matrix Estimation," Papers 1111.1133, arXiv.org, revised Mar 2013.
    3. Song Song, 2011. "Dynamic Large Spatial Covariance Matrix Estimation in Application to Semiparametric Model Construction via Variable Clustering: the SCE approach," Papers 1106.3921, arXiv.org, revised Jun 2011.
    4. Daye, Z. John & Jeng, X. Jessie, 2009. "Shrinkage and model selection with correlated variables via weighted fusion," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1284-1298, February.
    5. Clifford Lam & Jianqing Fan, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
    6. Chen, Ziqi & Shi, Ning-Zhong & Gao, Wei & Tang, Man-Lai, 2011. "Efficient semiparametric estimation via Cholesky decomposition for longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3344-3354, December.
    7. Clifford Lam, 2008. "Estimation of large precision matrices through block penalization," LSE Research Online Documents on Economics 31543, London School of Economics and Political Science, LSE Library.
    8. Peter Bickel & Bo Li & Alexandre Tsybakov & Sara Geer & Bin Yu & Teófilo Valdés & Carlos Rivero & Jianqing Fan & Aad Vaart, 2006. "Regularization in statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 15(2), pages 271-344, September.
    9. Zvi Bodie & J�r�me Detemple & Marcel Rindisbacher, 2009. "Life-Cycle Finance and the Design of Pension Plans," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 249-286, November.
    10. Xueying Zheng & Wing Fung & Zhongyi Zhu, 2013. "Robust estimation in joint mean–covariance regression model for longitudinal data," Annals of the Institute of Statistical Mathematics, Springer, vol. 65(4), pages 617-638, August.
    11. Mao, Jie & Zhu, Zhongyi & Fung, Wing K., 2011. "Joint estimation of mean-covariance model for longitudinal data with basis function approximations," Computational Statistics & Data Analysis, Elsevier, vol. 55(2), pages 983-992, February.
    12. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.

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