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Sparse semiparametric discriminant analysis

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  • Mai, Qing
  • Zou, Hui

Abstract

In recent years, a considerable amount of work has been devoted to generalizing linear discriminant analysis to overcome its incompetence for high-dimensional classification (Witten and Tibshirani, 2011, Cai and Liu, 2011, Mai et al., 2012 and Fan et al., 2012). In this paper, we develop high-dimensional sparse semiparametric discriminant analysis (SSDA) that generalizes the normal-theory discriminant analysis in two ways: it relaxes the Gaussian assumptions and can handle ultra-high dimensional classification problems. If the underlying Bayes rule is sparse, SSDA can estimate the Bayes rule and select the true features simultaneously with overwhelming probability, as long as the logarithm of dimension grows slower than the cube root of sample size. Simulated and real examples are used to demonstrate the finite sample performance of SSDA. At the core of the theory is a new exponential concentration bound for semiparametric Gaussian copulas, which is of independent interest.

Suggested Citation

  • Mai, Qing & Zou, Hui, 2015. "Sparse semiparametric discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 175-188.
    • Handle: RePEc:eee:jmvana:v:135:y:2015:i:c:p:175-188
    DOI: 10.1016/j.jmva.2014.12.009
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    References listed on IDEAS

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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Chen, Xiaohong & Fan, Yanqin, 2006. "Estimation of copula-based semiparametric time series models," Journal of Econometrics, Elsevier, vol. 130(2), pages 307-335, February.
    3. Chen, Xiaohong & Fan, Yanqin & Tsyrennikov, Viktor, 2006. "Efficient Estimation of Semiparametric Multivariate Copula Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1228-1240, September.
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    5. Peter Xue‐Kun Song, 2000. "Multivariate Dispersion Models Generated From Gaussian Copula," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(2), pages 305-320, June.
    6. Trendafilov, Nickolay T. & Jolliffe, Ian T., 2007. "DALASS: Variable selection in discriminant analysis via the LASSO," Computational Statistics & Data Analysis, Elsevier, vol. 51(8), pages 3718-3736, May.
    7. Qing Mai & Hui Zou & Ming Yuan, 2012. "A direct approach to sparse discriminant analysis in ultra-high dimensions," Biometrika, Biometrika Trust, vol. 99(1), pages 29-42.
    8. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
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    10. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
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    2. Jianqing Fan & Lingzhou Xue & Hui Zou, 2016. "Multitask Quantile Regression Under the Transnormal Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1726-1735, October.

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