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

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 low level of dependence. This is especially relevant in finance and insurance applications when data is 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 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.