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Efficient estimation in the bivariate normal copula model: Normal margins are least favourable. (English) Zbl 0877.62055
Summary: Consider semi-parametric bivariate copula models in which the family of copula functions is parametrized by a Euclidean parameter \(\theta\) of interest and in which the two unknown marginal distributions are the (infinite-dimensional) nuisance parameters. The efficient score for \(\theta\) can be characterized in terms of the solutions of two coupled Sturm-Liouville equations. Where the family of copula functions corresponds to the normal distributions with mean 0, variance 1 and correlation \(\theta\), the solution of these equations is given, and we thereby show that the normal scores rank correlation coefficient is asymptotically efficient. We also show that the bivariate normal model with equal variances constitutes the least favourable parametric submodel. Finally, we discuss the interpretation of \(|\theta |\) in the normal copula model as the maximum (monotone) correlation coefficient.

MSC:
62H12 Estimation in multivariate analysis
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H10 Multivariate distribution of statistics
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