Estimation

In typical settings, a copula model for a random vector \(X = (X_1,\dots , X_d)\) arises when the copula \(C(\cdot )\) of X is unknown but assumed to belong to a class \(\displaystyle \mathcal {C}_0 = \{C_\theta ~|~ \theta \in \mathcal {O}\}, \) where \(\mathcal {O}\) is an open subset of \(\mathbb {R}^p\) for some integer \(p \geq 1\) . When \(C_\theta (\cdot )\) is supposed to be a d-HAC \(C_{(\mathcal {V},\mathcal {E},\Psi )}\) in the sense of Definition 3.2 , the parameter (vector) \(\theta \) could be represented by the triplet \((\mathcal {V},\mathcal {E},\Psi )\) , which fully determines the possible HAC model. In practice, this model typically contains the following three ingredients: It is necessary to realize that all three ingredients are interconnected. Without knowing the Archimedean family, the parameters cannot be estimated, and without knowing the structure, the model cannot even be defined and thus neither the family nor the parameters can be obtained. This chapter attempts to present an answer to the following question: Under the assumption \(\displaystyle \begin{aligned} H_0 : C \in \mathcal{C}_0, \end{aligned}\) where \(\mathcal {C}_0\) is a set of d-HACs parametrized by those three ingredients, how these parameters could be estimated given a random sample \((x_{11},\dots ,x_{1d}), \dots , (x_{n1},\dots , x_{nd})\) drawn from \((X_1,\dots , X_d)\) ? In the literature, we often find approaches that focus just on a single ingredient. These then either assume the remaining ingredients to be known a priori (for example, using the dominant or expert knowledge) or circumvent their necessity. For example, Segers and Uyttendaele (2014), Matsypura et al. (2016), and Uyttendaele (2018) focus purely on the structure ingredient, whereas Savu and Trede (2010) focus on the parameter ingredient. Knowledge acquired by these researchers opens important doors for further investigations: In the case of Savu and Trede (2010), we obtain the best possible estimators in the sense of maximum likelihood. Thus, no other parameter estimator can be better, in terms of consistency and efficiency, in fixed samples while simultaneously estimating structure and parameters. Using knowledge from Segers and Uyttendaele (2014), Matsypura et al. (2016), and Uyttendaele (2018), we obtain the structure that can be used in estimating parameters following Savu and Trede (2010). However, this two-step procedure contains some pitfalls since errors made on one of the steps are propagated to the other. Another branch of approaches tries to estimate all three ingredients at once (estimation of the structure and parameters with the selection of the AC family), see Górecki et al. (2017b), or simplify the task by fixing one family for all nested ACs and estimate the remaining two ingredients, see Okhrin et al. (2013b), Górecki et al. (2016, 2021), and Cossette et al. (2019). In the rest of this chapter, we recall these approaches applied to each of the ingredients one by one. Before we do so, we briefly describe a general framework for all estimation approaches.