copulaCdfFrank

copulaCdfFrank(alpha, X)

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

alpha is a DOUBLE scalar that specifies the Frank copula shape parameter θ. The valid range is (-∞, ∞) \ {0}.

• alpha > 0 indicates positive correlation.

• alpha < 0 indicates negative correlation.

• A larger absolute value of alpha indicates stronger correlation.

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 Frank 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 Frank copula with shape parameter alpha=2.0 at the center point [0.5, 0.5].

X = matrix([0.5], [0.5])
y = copulaCdfFrank(2, X)
y
// Output: [0.3100572534791388]

Example 2. Calculate the cumulative probability of the Frank 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 = copulaCdfFrank(5.0, X)
y
// Output: [0.010103142862679,0.377148510746521,0.910103142862679,0.09942984775787]

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 = copulaCdfFrank(5.0, X)
y
// Output: [0,0.136404530970596,1]

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

X = copulaRandFrank(5.0, 1000)
fitRes = copulaFitFrank(X)
y = copulaCdfFrank(fitRes.alpha, X)
avg(y)
// Output: 0.35775959247079653

Related Functions: copulaFitFrank, copulaRandFrank, copulaPdfFrank, copulaCdfGaussian, copulaCdfStudent, copulaCdfClayton, copulaCdfGumbel