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Which copula estimation approach performs better when the empirical data to be modeled has a small sample size?

  • Parametric copulas (Gaussian, t-, Gumbel, Clayton, etc), or
  • Non-parametric (empirical) copula: histogram, kernel density based approaches that involve binning the samples

What is the nature of the problem that small samples impose on accurate copula estimation? Do the problems of whether parametric or non-parametric (marginal) density estimators spillover to copula estimators?

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The bias-variance trade-off suggests smaller samples will favor parametric approaches. Nonparametric models have high variance, and that becomes highly problematic in small samples. Parametric models have low variance and therefore are more usable in small samples. (The opposite holds in large samples.) Copula estimation is just a specific instance of this general law, and so a parametric copula is likely to do better than a nonparametric approach given a small sample.

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  • $\begingroup$ Thanks for confirming parametric copula lend themselves better to small sample data. If the parametric copula still shows itself to be inaccurate, however, how else can joint distributions and their dependence structure be modeled, without copula altogether? Or if sticking with copula, for the simple lack of an alternative, would the calculation (approximation) of the copula density by Monte Carlo integration improve accuracy in small sample settings? $\endgroup$
    – develarist
    Commented Sep 29, 2020 at 19:56
  • $\begingroup$ @develarist, tough questions. I cannot really comment on Monte Carlo integration; I would need some more details on how exactly that is defined and implemented. Regarding alternatives to copula models in small samples, we would be looking for something less flexible to avoid overfitting that is of particular concern. A multivariate $t$-distribution could perhaps be considered, with the covariance matrix estimated using some form of regularization, e.g. Bayesian (frequentist would usually require tuning the regularization intensity, and that would again be prone to overfitting). $\endgroup$
    – Richard Hardy
    Commented Sep 29, 2020 at 20:18

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