Copulas
Copulas are multivariate probability distributions with uniform marginals. They are useful because, due to Sklar’s theorem, they break up the specification of a joint distribution into two independent pieces: specification of the marginal distributions and specification of the dependence relation.
Literature
Books and surveys
- Schmidt, 2007: Coping with copulas (pdf)
- A very nice introduction to the basic concepts
- Joe, 1997: Multivariate models and dependence concepts (doi)
- Nelsen, 2006: An introduction to copulas, 2nd ed. (doi)
- The standard reference; dull but useful as a reference
- Durante & Sempi, 2015: Principles of copula theory (doi)
- Gives three different proofs of Sklar’s theorem, plus references to more
Triangular norms
Copulas are related to triangular norms (aka t-norms ) and triangular conorms (aka t-conorms or s-norms). In the literature, triangular norms are associated with fuzzy logic, a subject of which I am skeptical. As Dennis Lindley memorably put it, “anything fuzzy logic can do, probability can do better.” However, triangular norms are still interesting due to their connection to copulas and even in their own right, as ordered commutative monoids on the unit interval.
- Schweizer & Sklar, 1983: Probabilistic metric spaces, Ch. 5: Associativity & Ch. 6: Copulas
- Klement, Mesiar, Pap, 2000: Triangular norms (doi)
- Alsina, Frank, Schweizer, 2006: Associative functions: Triangular norms and copulas (doi)
See also functional equations.
Miscellaneous papers