An estimator of the stable tail dependence function based on the empirical beta copula (Q1633435)

This paper is based on a very interesting smoothed version of the well known empirical copula. Let \((X_{i1},\dots,X_{id})\), \(1\leq i\leq n\), be independent and identically distributed random vectors (rvs) with common continuous distribution function (df) \(F\). Denote by \(F_j\) the (continuous) df of \(X_{1j}\), \(1\leq j\leq d\). The rvs \((F_1(X_{i1}),\dots,F_d(X_{id}))\), \(1\leq i\leq n\), then follow that copula which corresponds to \(F\), say \(C\). Let \(\hat F_{nj}\) denote the usual empirical df of the univariate sample \(X_{1j},\dots,X_{nj}\), \(1\leq j\leq d\). The empirical df of the sample of \(d\)-variate rvs \(\left(\hat F_{n1}(X_{i1}),\dots,\hat F_{nd}(X_{id})\right)\), \(1\leq i\leq n\), is the empirical copula, say \(C_n\), i.e., \(C_n(u)= n^{-1}\sum_{i=1}^n\prod_{j=1}^d 1\left(\hat F_{nj}(X_{ij})\leq u_j \right)\), \(u=(u_1,\dots,u_d)\in[0,1]^d\). Given that \(\hat F_{nj}(X_{ij})=r/n\), the event \(\{\hat F_{nj}(X_{ij})\leq u_j\}\) can approximately be interpreted as the event that the \(r\)-th order statistic \(U_{r:n}\) in a sample of independent and uniformly on \([0,1]\) distributed random variables satisfies \(U_{r:n}\leq u_j\). It is reasonable to predict the random variable \(1(U_{r:n}\leq u_j)\) by its expectation, which is \(P(U_{r:n}\leq u_j)=\sum_{s=r}^n\binom{n}{s}u_j^s(1-u_j)^{n-s}=:F_{nr}(u_j)\). Doing this simultaneously for each factor \(1\left(\hat F_{nj}(X_{ij})\leq u_j \right)\) in \(C_n(u)\), provides the definition of the empirical beta copula \(C_n^\beta(u)=n^{-1}\sum_{i=1}^n\prod_{j=1}^d F_{nR_{ij}}(u_j)\), where \(R_{ij}= n \hat F_{nj}(X_{ij})\). Different to \(C_n\), \(C_n^\beta\) is actually a copula. The authors use \(C_n^\beta\) to estimate the stable tail dependence function corresponding to \(C\). This estimator has the same limit behaviour as that based on \(C_n\), but its finite sample behaviour is superior. The empirical beta copula enables also a simple and effective resampling scheme.