emp.copula: Empirical copula

Description

emp.copula and emp.copula.self compute the empirical copula for a given sample. The difference between these functions is, that emp.copula.self takes the sample itself as grid u, such that the grid has not to be specified manually.

Usage

emp.copula(u, x, proc = "M", sort = "none", margins = NULL, 
na.rm = FALSE, max.min = TRUE)
emp.copula.self(x, proc = "M", sort = "none", margins = NULL, 
na.rm = FALSE, max.min = TRUE)

Arguments

the grid for the computation of the empirical copula. It can be a scalar, a vector or a matrix. The number of columns has to be equal to the number of columns of the matrix x. The values of u should be within the interval $[0, 1]

denotes the matrix of variables, which are marginal distributions in practice. The values should be within the interval $[0, 1]$. The number of columns should correspond to the number of variables, whereas the number of rows should be equal to the number

proc

enables the user to choose between two different methods. Generally, it is advised to choose "M" (given by default), because it takes only $1/25$ of the computational time of method "A". However, in the case of large datasets it

sort

defines, whether the output is ordered. sort = "asc" refers to ascending values, which might be interesting for plotting and sort = "desc" refers to descending values.

margins

scalar or vector specifying how the margins are to compute. They can be determined nonparametrically denoted by "edf" or in parametric way , e.g. "norm". See estimate.copula

na.rm

boolean. If na.rm = TRUE and a row of x or u contains NA, the row is removed and not used for the computation. Corresponding warnings are shown.

max.min

boolean. If max.min = TRUE and an element of x or u is $> 1$ or $< 0$, it is set to $1$ and $0$ respectively.

Value

A vector containing the values of the empirical copula.

Details

The estimated copula follows the formula $$\widehat{C} \left(u_{1}, \dots, u_{d} \right) = n^{-1} \sum_{i=1}^{n} \prod_{j=1}^{d} \mathbf{I} \left{ \widehat{F}_{j} \left( X_{ij} \right) \leq u_{j} \right},$$ where $\widehat{F}_{j}$ denotes the empirical marginal distribution function of variable j. So it is no smoothing implemented yet.

Examples

Run this code

v = seq(-4, 4, 0.005)
X = cbind(matrix(pt(v, 1), 1601, 1), matrix(pnorm(v), 1601, 1))

# both methods lead to the same result 
z = emp.copula.self(X, proc = "M") 
which(((emp.copula.self(X[1:100, ], proc = "M") - emp.copula.self(X[1:100, ],
proc = "A")) == 0) == "FALSE")
# integer(0)

# furthermore recognize the contour plot
out = outer(z, z)
contour(x = X[,1], y = X[,2], out, main = "Contour Plot", 
xlab = "Cauchy Margin", ylab = "Standard Normal Margin", 
labcex = 1, lwd = 1.5, nlevels = 15)