copulaCdfGumbel

copulaCdfGumbel(alpha, X)

Calculates the cumulative probability of the Gumbel copula, with scalar parameter alpha evaluated at the points in X.

alpha is a DOUBLE scalar that specifies the Gumbel copula shape parameter θ. The valid range is [1, ∞).

• alpha = 1 indicates an independent copula.

• alpha > 1 indicates positive dependence and upper-tail dependence.

• A larger alpha indicates stronger upper-tail dependence.

X is a non-empty two-dimensional numeric matrix or table with dimensions n×p (only p = 2 is currently supported), specifying the set of evaluation points for which densities are computed. Here, n is the number of evaluation points, that is, the number of rows in X; p is the number of variables, that is, the number of columns in X.

• All elements must be finite numbers.

• Rows whose elements are all in the open interval (0, 1) are evaluated using the Gumbel copula cumulative distribution formula.

• Rows with any element u <= 0 return 0. Rows with all elements u >= 1 return 1.

• When X is a table, each column represents a variable, and the column order defines the variable order.

A DOUBLE vector with the same length as the number of rows in X. The i-th element of the returned vector is the copula cumulative probability for the i-th row of X.

Example 1. Calculate the cumulative probability of the Gumbel copula with shape parameter alpha=2.0 at the center point [0.5, 0.5].

X = matrix([0.5], [0.5])
y = copulaCdfGumbel(2.0, X)
y
// Output: [0.37521422724648174]

Example 2. Calculate the cumulative probability of the Gumbel copula at multiple evaluation points.

u1 = [0.05, 0.5, 0.95, 0.1]
u2 = [0.05, 0.5, 0.95, 0.9]
X = matrix(u1, u2)
y = copulaCdfGumbel(2.0, X)
y
// Output: [0.0144565856995,0.375214227246482,0.930028849282898,0.099759364396173]

Example 3. Demonstrate how boundary points are handled.

u1 = [0.0, 0.2, 1.0]
u2 = [0.1, 0.3, 1.0]
X = matrix(u1, u2)
y = copulaCdfGumbel(2.0, X)
y
// Output: [0,0.133997310771081,1]

Example 4. Calculate the cumulative probability using the fitted Gumbel shape parameter.

X = copulaRandGumbel(2.0, 1000)
fitRes = copulaFitGumbel(X)
y = copulaCdfGumbel(fitRes.alpha, X)
avg(y)
// Output: 0.3880172112127464

Related Functions: copulaFitGumbel, copulaRandGumbel, copulaPdfGumbel, copulaCdfGaussian, copulaCdfStudent, copulaCdfClayton, copulaCdfFrank