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Maximum Pseudo-Likelihood Estimation of Copula Models and Moments of Order Statistics

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  • Alexandra Dias

    (School for Business and Society, University of York, York YO10 5DD, UK)

Abstract

It has been shown that, despite being consistent and in some cases efficient, maximum pseudo-likelihood (MPL) estimation for copula models overestimates the level of dependence, especially for small samples with a low level of dependence. This is especially relevant in finance and insurance applications when data are scarce. We show that the canonical MPL method uses the mean of order statistics, and we propose to use the median or the mode instead. We show that the MPL estimators proposed are consistent and asymptotically normal. In a simulation study, we compare the finite sample performance of the proposed estimators with that of the original MPL and the inversion method estimators based on Kendall’s tau and Spearman’s rho. In our results, the modified MPL estimators, especially the one based on the mode of the order statistics, have a better finite sample performance both in terms of bias and mean square error. An application to general insurance data shows that the level of dependence estimated between different products can vary substantially with the estimation method used.

Suggested Citation

  • Alexandra Dias, 2024. "Maximum Pseudo-Likelihood Estimation of Copula Models and Moments of Order Statistics," Risks, MDPI, vol. 12(1), pages 1-26, January.
    • Handle: RePEc:gam:jrisks:v:12:y:2024:i:1:p:15-:d:1321870
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    References listed on IDEAS

    as
    1. Nasri, Bouchra R. & Rémillard, Bruno N. & Bouezmarni, Taoufik, 2019. "Semi-parametric copula-based models under non-stationarity," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 347-365.
    2. Genest, Christian & Segers, Johan, 2010. "On the covariance of the asymptotic empirical copula process," LIDAM Reprints ISBA 2010017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Genest, Christian & Segers, Johan, 2010. "On the covariance of the asymptotic empirical copula process," LIDAM Reprints ISBA 2010038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Joe, Harry, 2005. "Asymptotic efficiency of the two-stage estimation method for copula-based models," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 401-419, June.
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