Estimators based on trimmed Kendall’s tau in multivariate copula models

A common method of estimating the parameters of dependency in multivariate copula models is by maximum likelihood principle, termed as Inference From Marginals (IFM); see Joe (1997)  [13]. To avoid possible misspecification of the marginal distributions, some authors suggest rank-based procedures for estimating the parameters of dependency in a multivariate copula model. A standard approach for this problem is through maximization of the pseudolikelihood, as discussed in Genest et al. (1995)  [9] and Shih and Louis (1995)  [23]. Alternative estimators based on the inversion of two multivariate extensions of Kendall’s tau, due to Kendall and Babington Smith (1940)  [14] and Joe (1990)  [12], were used in Genest et al. (2011)  [10]. In the literature, dependency of data was considered in the whole data space. However, it may be better to divide the data set into two distinct sets, lower and higher than a threshold, and then evaluate the dependency parameters in these sets. In this way, we may have different dependency parameters in these sets which may shed additional light. For example, in drought analysis, precipitation and minimum temperature may be modeled using copulas in which case we can infer that dependency between precipitation and minimum temperature are severe when they are less than a certain threshold. In this paper, after introducing trimmed Kendall’s tau when such a threshold is imposed, we consider modeling dependency using it as a measure. Asymptotic distribution of trimmed Kendall’s tau is also investigated, and a test for the null hypothesis of equality between Kendall’s tau and trimmed Kendall’s tau is constructed. We can use this hypothesis testing procedure for testing the hypothesis that data are dependent before a threshold value and are independent after the threshold. An explicit form of the asymptotic distribution of trimmed Kendall’s tau and of the mentioned test statistic are also derived for some special families of copulas. Finally, the results of a simulation study and an illustrative example are provided.