1.1 Definition viacdf

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

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 canintroduce an interpolating parameter\(\theta\geq 0\) such that\(\theta =0\) results in the independence copula, and\(\theta = \infty\) results in the DJ copulaclass – henceinterpolating DJ copulas with the independencecopula.

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\[H_{\psi}(x; \eta) = e ^ {-x} \psi(\etax)\] for\(x > 0\) and\(\eta \geq 0\), and has a range between 0and 1. Its derivative (with respect to the first argument) is alsouseful:\[\text{D}_1 H_{\psi}(x; \eta) = - e^ {-x} (\psi(\eta x) - \eta \psi'(\eta x)).\]

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

All formulas in this vignette can be found inCoia (2017), but under a different (and morecomplex) 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 slightlydifferent.

1.2 Kappa Transform of\(\psi\)

Besides\(H_{\psi}\), anothertransformation of\(\psi\) is necessaryfor 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 DJcopulas – the requirement has to do with the\(\kappa\) function. So, in case you want togo in the opposite direction, and obtain\(\psi\) from a choice of\(\kappa\), we can do so by solving thedifferential 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: DJcopula 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 byconditioning 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 functionhas 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 fromCoia (2017) Section E.1.2, note that theformulas 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 byconditioning on either the 1st or 2nd variable, have a convenient formif 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 functionhas 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\(\frac{\text{d}y}{\text{d}v} = - 1 / \text{D}_1H_\psi(y; \theta)\).

2.1 GeneratingFunctions

The generating functions of the IG and IGL copula families rely onthe 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) incompleteGamma function defined as\[\Gamma^{*}(\alpha, x) = \int_x^{\infty} t ^{\alpha - 1} e ^ {-t} \text{d} t.\] The density function isdenoted by\(f_{\alpha}\).

To emphasize the dependence of\(\psi\),\(\kappa\), and\(H\) on the parameter\(\alpha\), this parameter is written as asubscript. The\(\psi\) function andits 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 forvarious values of\(\alpha\).

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

The\(H\) function does notsimplify, although for convenience,\(H_{\psi_{\alpha}}\) is simply denoted\(H_{\alpha}\). Here are some plots of thesefunctions, compared with the plot of the negative exponential\(e^{-x}\) as the faded dotted line. Noticethat increasing\(\alpha\) draws\(H_{\alpha}(\cdot, \eta)\) closer to thenegative exponential, whereas increasing\(\eta\) pulls it further.

2.2 CopulaQuantities

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

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