Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas
Marten Wegkamp, Yue Zhao(Submitted on 28 May 2013 (v1), last revised 6 Jan 2014 (this version, v2))
We study the adaptive estimation of copula correlation matrix Σ for elliptical copulas. In this context, the correlations are connected to Kendall's tau through a sine function transformation. Hence, a natural estimate for Σ is the plug-in estimator Σˆ with Kendall's tau statistic. We first obtain a sharp bound for the operator norm of Σˆ−Σ. Then, we study a factor model for Σ, for which we propose a refined estimatorΣ˜ by fitting a low-rank matrix plus a diagonal matrix to Σˆ using least squares with a nuclear norm penalty on the low-rank matrix. The bound for the operator norm of Σˆ−Σ serves to scale the penalty term, and we obtain finite sample oracle inequalities for Σ˜. We also consider an elementary factor model of Σ, for which we propose closed-form estimators. We provide data-driven versions for all our estimation procedures and performance bounds.
Subjects: Machine Learning (stat.ML)Cite as: arXiv:1305.6526 [stat.ML] (or arXiv:1305.6526v2 [stat.ML] for this version)