I'm reading the paper by Lourme A. et al (2016) and tried to plot $2D$ confidence areas like on the Fig.2. Discussion about confidence areas of normal distributed data and some results are here.

Let's remind the definition of confidence areas from the paper:

4.1.2. Confidence areas

Let $u = (u_1,…,u_d)′$ be a random vector with uniform margins on $(0,1)$ and let $y = (y_1,…, y_d)′ \in \mathbb{R}$ be defined by: $y_j = G(u_j), ( j \in {1,…,d})$ where $G$ is a continuous increasing map defined between $0$ and $1$.

Let us assume that the multivariate c.d.f of $u$ is a Gaussian copula with parameter $R_g$. Taking the quantile function of $\mathcal{N_1}(0, 1)$ as $G$, then $y$ is distributed as $\mathcal{N_d}(0, R_g)$ and the random variable $y′R_g^{−1} y$ as $χ^2_d$. Given $\alpha \in (0, 1)$, the latter variable $y′R_g^{−1} y$ is less than $χ_{d,α}^2$ – the $\alpha$ order quantile of $χ^2_d$ – with probability $\alpha$. Hence, $$\Gamma_g(\alpha)=\{\nu=(\nu_1,\ldots, \nu_d)' \in [0, 1]^d; \\(G(\nu_1), \ldots,G(\nu_d)) R_g^{-1}(G(\nu_1), \ldots,G(\nu_d))' \le \chi^2_{d,\alpha}\}$$ is a $d$-dimensional compact confidence area inside (resp. outside) of which $u$ falls with probability $\alpha$ (resp. $1 − \alpha$).

Also in the paper one can find the formula of $d$-dimensional compact confidence area for Student's t distributed data:

$$\Gamma_t(\alpha)=\{\nu=(\nu_1,\ldots, \nu_d)' \in [0, 1]^d; \\(G(\nu_1), \ldots,G(\nu_d)) R_t^{-1}(G(\nu_1), \ldots,G(\nu_d))' \le d \times \mathcal{F}_{d, v, \alpha}\}$$

where $\mathcal{F}_{d, v, \alpha}$ – the $\alpha$ order quantile of the Fisher distribution $\mathcal{F}_{d, \nu}$ – with probability $\alpha$, and $d$ and $v$ degrees of freedom.

Trus, the $d$-dimensional compact confidence area is defind for the elliptical case (Normal and Student's $t$).

Question. Could someone explain a used approach in order to derive a confidence area formula to multivariate distibuted data that approximated with an Archimedean copula? for instance, a Frank's copula or a Gumbel one.

Edit 2.

Lets $\Psi_\infty$ is the class of completely monotone generators $\psi$, i.e., $(−1)^k\psi^{(k)}(t) \ge 0$ for all $k \in \mathbb{N}_0$. By Bernstein’s Theorem, the class of completely monotone generators precisely coincides with the class of Laplace–Stieltjes transforms ($\mathcal{LS}$) of distribution functions $F$ on the positive real line, that is, any completely monotone generator allows for the representation Grothe et al (2014) $$\psi(t) = \mathcal{LS}[F](t) = \int^{\infty}_0 exp(−tx) dF(x).$$ This representation has an important consequence for sampling since it provides a stochastic representation for $U \sim C$. In the paper by Hofert et al (2012) one can found

A random vector $U$ following an Archimedean copula with generator $\psi ∈ \Psi_\infty$ allows for the stochastic representation $U=(\psi(E_1/V),...,\psi (E_d/V))^⊤$, where $V \sim F=\mathcal{LS}^{−1}(\psi)$ and i.i.d. $E_j \sim Exp(1)$, $j \in \{1,..., d\}$.

$\mathcal{LS}^{−1}(\psi)$ is the inverse of Laplace–Stieltjes transform of a generator $\psi$, for the Clayton copula the parameter $\theta \in (0,\infty)$, the generator $\psi(t)= (1+t)^{−1/\theta}$, and $ V \sim \Gamma(1/\theta,1)$.

Archimedean families (Ali-Mikhail-Haq, Clayton, Frank, Gumbel, and Joe) with corresponding parameter ranges, generators, and inverse Laplace-Stieltjes transforms can be found in the paper Hofert (2008).

Reference. Lourme, A., F. Maurer (2016) Testing the Gaussian and Student's t copulas in a risk management framework. Economic Modelling.

Marius Hofert, Martin Mächler, Alexander J. McNeil. Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis 110 (2012) 133–150.

Marius Hofert (2008) Sampling Archimedean copulas. Part of the Hofert’s dissertation.

O. Grothe, M. Hofert, Construction and sampling of Archimedean and nested Archimedean L´evy copulas, Journal of Multivariate Analysis (2014), http://dx.doi.org/10.1016/j.jmva.2014.12.004


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