The Interdependence between Rainfall and Temperature: Copula Analyses

Table 4

Five families of copulas.

Family	𝐶(𝑢,𝑣)	Range of 𝜃
Normal	𝑁𝜃(Φ^−1(𝑢),Φ^−1(𝑣))	[−1, 1]
Student	𝑇𝜃,𝛾(𝑇𝛾^−1(𝑢),𝑇𝛾^−1(𝑣))	[−1, 1]
Clayton	(𝑢^−𝜃+𝑣^−𝜃−1)^−1/𝜃	(0,∞)
Frank	−𝜃^−1ln1+(𝑒^−𝜃𝑢−1)(𝑒^−𝜃𝑣−1)/(𝑒^−𝜃−1)	(−∞,∞)
Gumbel	exp−[(−ln𝑢)^𝜃+(−ln𝑣)^𝜃]^1/𝜃	[1,∞)

Φ: cumulative distribution function (CDF) of a 𝑁(0,1).
𝑁_𝜃: CDF of a standard bivariate normal distribution with Pearson
correlation 𝜃.
𝑇_𝛾: CDF of a student distribution with 𝛾 degrees of freedom.
𝑇_𝜃,𝛾: CDF of a bivariate student distribution with 𝛾 degrees of freedom.
Source: [46].