Biconic semi-copulas with a given section

Inspired by the notion of biconic semi-copulas, we introduce biconic semi-copulas with a given section. Such semi-copulas are constructed by linear interpolation on segments connecting the graph of a continuous and decreasing function to the points (0, 0) and (1, 1). Special classes of biconic semi-copulas with a given section such as biconic (quasi-)copulas with a given section are considered. Some examples are also provided

Focal copulas: a common framework for various classes of semilinear copulas

A new method to construct semi-copulas is introduced. These semi-copulas are called focal semi-copulas and their construction is based on linear interpolation on segments connecting the diagonal of the unit square with two focal points. Several classes of semilinear semi-copulas, such as lower semilinear semi-copulas, upper semilinear semi-copulas, ortholinear semi-copulas and biconic semi-copulas with a given diagonal section, turn out to be special cases of focal semi-copulas. Subclasses of focal semi-copulas, such as focal (quasi-)copulas are characterized as well

Orbital semilinear copulas

summary:We introduce four families of semilinear copulas (i.e. copulas that are linear in at least one coordinate of any point of the unit square) of which the diagonal and opposite diagonal sections are given functions. For each of these families, we provide necessary and sufficient conditions under which given diagonal and opposite diagonal functions can be the diagonal and opposite diagonal sections of a semilinear copula belonging to that family. We focus particular attention on the family of orbital semilinear copulas, which are obtained by linear interpolation on segments connecting the diagonal and opposite diagonal of the unit square

Semiquadratic copulas based on horizontal and vertical interpolation

We introduce several families of semiquadratic copulas (i.e. copulas that are quadratic in any point of the unit square in at least one coordinate) of which the diagonal and/or opposite diagonal sections are given functions. These copulas are constructed by quadratic interpolation on segments connecting the diagonal, opposite diagonal and sides of the unit square; all interpolations are therefore performed horizontally or vertically. For each family we provide the necessary and sufficient conditions on the given diagonal and/or opposite diagonal functions and two auxiliary real functions to obtain a copula that has these diagonal and/or opposite diagonal functions as diagonal and/or opposite diagonal sections. Just as the product copula is a central member of all families of semilinear copulas based on horizontal and vertical interpolation, it turns out that the Farlie Gumbel Morgenstern family of copulas is included in all families of semiquadratic copulas introduced and characterized here