1.1 Definition via cdf

Both the DJ and the Interpolated DJ copula classes (and thus the IG and IGL copulas) are characterized by a generating function \(\psi:[0, \infty) \rightarrow (0, 1]\), which is a concave distribution function or convex survival function.

A DJ copula (and thus an IGL copula family) has cdf \[C_{\text{DJ}}(u, v; \psi) = u + v - 1 + (1 - u) \psi\left((1 - u) \psi^{\leftarrow}(1 - v)\right),\] for \((u, v)\) in the unit square, where \(\psi^{\leftarrow}\) is the left-inverse of \(\psi\).

But, this class does not contain the independence copula, so we can introduce an interpolating parameter \(\theta \geq 0\) such that \(\theta = 0\) results in the independence copula, and \(\theta = \infty\) results in the DJ copula class – hence interpolating DJ copulas with the independence copula.

An interpolated DJ copula (and thus an IG copula) has cdf \[C_{\text{intDJ}}(u, v; \theta, \psi) = u + v - 1 + (1 - u) H_{\psi}\left(H_{\psi}^{\leftarrow}(1 - v; \theta); (1 - u) \theta \right)\] for \((u, v)\) in the unit square, where \(H_{\psi}\) is the interpolating function of \(\psi\), defined as \[H_{\psi}(x; \eta) = e ^ {-x} \psi(\eta x)\] for \(x > 0\) and \(\eta \geq 0\), and has a range between 0 and 1. Its derivative (with respect to the first argument) is also useful: \[\text{D}_1 H_{\psi}(x; \eta) = - e ^ {-x} (\psi(\eta x) - \eta \psi'(\eta x)).\]

Although the DJ copula class can be considered part of the Interpolated DJ class if we include \(\theta = \infty\) in the parameter space, the formulas when \(\theta = \infty\) do not simplify in an obvious way, so it’s best to treat the two classes separately.

All formulas in this vignette can be found in Coia (2017), but under a different (and more complex) parameterization: the \(\psi\) function is \(\psi(1/x)\) and the \(H\) function is \(H_{\psi}(x; \theta) = \frac{1}{x} \psi(1 / (\theta \log x))\), and the copula formulas are correspondingly slightly different.

1.2 Kappa Transform of \(\psi\)

Besides \(H_{\psi}\), another transformation of \(\psi\) is necessary for writing formulas succinctly: \[\kappa_{\psi}(x) = \frac{\text{d}}{\text{d}x} x \psi(x) = \psi(x) + x \psi'(x)\] for \(x > 0\).

By the way, not all choices of \(\psi\) result in valid DJ / Interpolated DJ copulas – the requirement has to do with the \(\kappa\) function. So, in case you want to go in the opposite direction, and obtain \(\psi\) from a choice of \(\kappa\), we can do so by solving the differential equation in the definition of \(\kappa\), to get \[\psi(x) = \frac{1}{x} \int_{0}^{x} \kappa(t) \text{d}t.\]

1.3 Copula Quantities: DJ copula class (IGL copula family)

To simplify formulas, denote \(y = \psi^{\leftarrow}(1 - v)\). Throughout, \(p\), \(u\), and \(v\) are numbers between 0 and 1.

The density of a DJ copula (and thus an IGL copula) is \[c_{\text{DJ}}(u, v; \psi) = (1 - u) \frac{\kappa_{\psi}'((1 - u) y)}{\psi'(y)}\]

The two families of conditional distributions, obtained by conditioning on either the 1st or 2nd variable, are \[\begin{equation} \begin{split} C_{\text{DJ}, 2 | 1}(v | u; \psi) & = 1 - \kappa_{\psi}((1 - u) y); \\ C_{\text{DJ}, 1 | 2}(u | v; \psi) & = 1 - (1 - u) ^ 2 \frac{\psi'((1 - u) y)}{\psi'(y)}. \end{split} \end{equation}\] The corresponding quantile functions are the (left) inverses of these functions, for which the 2|1 quantile function has a closed form: \[C_{\text{DJ}, 2 | 1}^{-1}(p | u; \psi) = 1 - \psi\left((1 - u) ^ {-1} \kappa_{\psi}^{\leftarrow}(1 - p) \right)\]

To check these equations, note that \(\frac{\text{d}y}{\text{d}v} = - 1 / \psi'(y)\). If comparing to the formulas from Coia (2017) Section E.1.2, note that the formulas there are defined for the copula reflections.

1.4 Copula Quantities: Interpolated DJ copula class (IG copula family)

To simplify formulas, denote \(y = H_{\psi} ^ {\leftarrow}(1 - v; \theta)\). Throughout, \(p\), \(u\), and \(v\) are numbers between 0 and 1.

The density of an interpolated DJ copula (and thus an IG copula) is \[\begin{equation} c_{\text{intDJ}}(u, v; \theta, \psi) = \frac{\text{D}_1 H_{\kappa_{\psi}}(y; (1 - u) \theta)} {\text{D}_1 H_{\psi}(y; \theta)} \end{equation}\]

The two families of conditional distributions, obtained by conditioning on either the 1st or 2nd variable, have a convenient form if we consider the interpolating function \(H\) of the \(\kappa\) function instead of the \(\psi\) function: \[\begin{equation} \begin{split} C_{\text{intDJ}, 2 | 1}(v | u; \theta, \psi) & = 1 - H_{\kappa_{\psi}}\left(y; (1 - u) \theta \right) \\ C_{\text{intDJ}, 1 | 2}(u | v; \theta, \psi) & = 1 - (1 - u) \frac{\text{D}_1 H_{\psi}(y; (1 - u) \theta)} {\text{D}_1 H_{\psi}(y; \theta)}\\ \end{split} \end{equation}\] The corresponding quantile functions are the (left) inverses of these functions, for which the 2|1 quantile function has a closed form: \[\begin{equation} \begin{split} C_{\text{intDJ}, 2 | 1}^{-1}(p | u; \theta, \psi) & = 1 - H_{\psi} \left( H_{\kappa_{\psi}} ^ {\leftarrow} (1 - p; (1 - u) \theta); \theta \right) \end{split} \end{equation}\]

To check these equations, note that \(\frac{\text{d}y}{\text{d}v} = - 1 / \text{D}_1 H_\psi(y; \theta)\).

2.1 Generating Functions

The generating functions of the IG and IGL copula families rely on the Gamma(\(\alpha\)) distribution, which has cdf \[F_{\alpha}(x) = \frac{\Gamma(\alpha) - \Gamma^{*}(\alpha, x)}{\Gamma(\alpha)}\] for \(\alpha > 0\) and \(x \ge 0\), where \(\Gamma\) is the Gamma function, and \(\Gamma^{*}\) is the (upper) incomplete Gamma function defined as \[\Gamma^{*}(\alpha, x) = \int_x^{\infty} t ^ {\alpha - 1} e ^ {-t} \text{d} t.\] The density function is denoted by \(f_{\alpha}\).

To emphasize the dependence of \(\psi\), \(\kappa\), and \(H\) on the parameter \(\alpha\), this parameter is written as a subscript. The \(\psi\) function and its derivative are \[\begin{equation} \begin{split} \psi_{\alpha}(x) & = 1 - F_{\alpha}(x) + \frac{\alpha}{x} F_{\alpha + 1}(x); \\ \psi_{\alpha}'(x) & = - \frac{\alpha}{x ^ 2} F_{\alpha + 1}(x). \end{split} \end{equation}\] Here are some plots of these functions for various values of \(\alpha\).

The \(\kappa\) function is the Gamma survival function, \[\kappa_{\alpha}(x) = 1 - F_{\alpha}(x).\]

The \(H\) function does not simplify, although for convenience, \(H_{\psi_{\alpha}}\) is simply denoted \(H_{\alpha}\). Here are some plots of these functions, compared with the plot of the negative exponential \(e^{-x}\) as the faded dotted line. Notice that increasing \(\alpha\) draws \(H_{\alpha}(\cdot, \eta)\) closer to the negative exponential, whereas increasing \(\eta\) pulls it further.

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2.2 Copula Quantities

Most of the formulas for copula quantities do not simplify nicely from their general forms. But, here are the ones that do.

For the IGL copula family: \[\begin{equation} \begin{split} C_{\text{IGL}, 1 | 2}(u | v; \theta, \alpha) & = 1 - \frac{F_{\alpha + 1}((1 - u) y)}{F_{\alpha + 1}(y)}; \\ C_{\text{IGL}, 1 | 2}^{-1}(p | v; \theta, \alpha) & = 1 - \frac{1}{y} F_{\alpha + 1}^{-1}\left((1 - p) F_{\alpha + 1}(y)\right). \end{split} \end{equation}\]