The Annals of Statistics

Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation

Johan Segers, Ramon van den Akker, and Bas J. M. Werker

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We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations. We indicate how our results can be extended to joint regression models.

Article information

Ann. Statist., Volume 42, Number 5 (2014), 1911-1940.

First available in Project Euclid: 11 September 2014

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties
Secondary: 62B15: Theory of statistical experiments 62H20: Measures of association (correlation, canonical correlation, etc.)

Adaptivity correlation matrix influence function quadratic form ranks score function tangent space


Segers, Johan; van den Akker, Ramon; Werker, Bas J. M. Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation. Ann. Statist. 42 (2014), no. 5, 1911--1940. doi:10.1214/14-AOS1244.

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Supplemental materials

  • Supplementary material: Supplement to the paper: “Semiparametric Gaussian copula models”. The supplement contains the proofs for the results in this paper as well as some additional figures for the Monte Carlo simulations reported in Section 5.