Maximum entropy copula with given diagonal section

Abstract

We consider copulas with a given diagonal section and compute the explicit density of the unique optimal copula which maximizes the entropy. In this sense, this copula is the least informative among the copulas with a given diagonal section. We give an explicit criterion on the diagonal section for the existence of the optimal copula and give a closed formula for its entropy. We also provide examples for some diagonal sections of usual bivariate copulas and illustrate the differences between them and the maximum entropy copula with the same diagonal section.

Maximum entropy distribution of order statistics with given marginals

“…. According to (6), we deduce that G −1 • G • F −1 i = F −1 i on (0, 1). This implies that for s, t ∈ (0, 1), 1 ≤ i < j ≤ d:…”

“…In particular, we obtain that J(δ) is equal to − log(2) + J (δ (2) ), with J as also defined by (1) in [6]. Therefore we deduce case (a) of Theorem 4.7 (for d = 2) from case (a) of Theorem 2.4 in [6]. Then, we get from (15) and Theorem 4.7 case (a) that H(F ) = −∞ for all F ∈ L OS 2 (F).…”

“…on Z δ F defined by (39). Then use (4) and (6) to get that δ −1 (i) • δ (i) (u (i) ) = u (i) a.e. on Z c δ F .…”

Coordinate-wise transformation of probability distributions to achieve a Stein-type identity

“…. , X d ) has a Stein-type distribution, then the sum j X j is positively correlated with any monotone transformation of X i due to (5). This property is applied to a rating problem in Sect.…”

“…where T = (T i ) is the Stein-type transformation of μ. The quantity g(x) satisfies E[g(X ) f (X i )] ≥ 0 for any increasing function f (x i ) of x i due to the Stein-type identity (5). In particular, by taking the step function f…”