ECBC {qad}

Calculate empirical checkerboard copula

Description

The function ECBC() computes the mass distribution of the empirical (checkerboard) copula, given a bi-variate sample. If resolution equals sample size, the bi-linearly extended empirical copula is returned. Note, if there are ties in the sample an adjusted empirical copula is calculated. The function ECBC.eval() evaluates the checkerboard copula at given points.

Usage

ECBC(X, Y, resolution = NULL)

ECBC.eval(CB, eval.points)

Arguments

X

Numeric vector of values in the first coordinate

Y

Numeric vector of values in the second coordinate

resolution

Integer indicating the resolution of the checkerboard aggregation, i.e. the number of vertical/horizontal strips of the checkerboard copula. (default = NULL, sets resolution = floor(sqrt(sample size)))

CB

A numeric mass matrix of a checkerboard copula (ECBC)

eval.points

A numeric matrix or data.frame indicating the eval.points (x,y)

Details

If the observations are drawn from a continuous distribution (no ties in the sample), the function ECBC() returns the commonly used empirical checkerboard copula. If there are ties in the sample, the empirical copula is adjusted and calculated in the following way:
Let (u_i,v_i) := (F_n(x_i),G_n(y_i)) be the pseudo-observations for i in {1,...,n} and (u_1',v_1'),..., (u_m',v_m') the distinct pairs of pseudo-observations with m leq n. Moreover set S_1:={0, u_1, ..., u_m_1} and S_2:={0, v_1,..., v_m_2} and define the quantities t_i,r_i and s_i for i=1,..., m by

t_i := sum_{j=1}^n 1_{(u_i',v_i')}(u_j,v_j)

r_i := sum_{j=1}^n 1_{u_i}(u_j)

s_i := sum_{j=1}^n 1_{v_i}(v_j)

where 1 defines the indicator function. Define the empirical subcopula A'_n: S_1 x S_2 to {0,1/n, ..., (n-1)/n,1} by

A'_n(s_1,s_2)= 1/n sum_{i=1}^m t_i * 1_{[0,s_1] x [0,s_2]} (u_i', v_i')=1/n sum_{i=1}^n 1_{[0,s_1] x [0,s_2]} (u_i, v_i)

for all s_1 in S_1 and s_2 in S_2.
We extend the subcopula A'_n to a copula by defining the transformations w_i:[0,1]^2 to [u_i'-r_i/n,u_i'] x [v_i'-s_i/n,v_i'] by

w_i(x,y)=(u_i'-r_i/n+r_i*x/n, v_i'-s_i/n + s_iy/n)

and set the measure of the empirical copula mu_A_n^B := 1/n sum_i=1^m t_i mu_B^w_i, where B denotes the product copula.

Value

ECBC() returns a matrix with the mass distribution of the empirical (checkerboard) copula.

References

Deheuvels, P. (1979). La fonction de dependance empirique et ses proprietas: un test non parametrique d'independance, Acad. Roy. Belg. Bull. Cl. Sci., 5th Ser. 65, 274-292.

Li, X., Mikusinski, P. and Taylor, M.D. (1998). Strong approximation of copulas, Journal of Mathematical Analysis and Applications, 255, 608-623.

Genest, C., Neshlehova J.G. and Remillard, B. (2014). On the empirical multilinear copula process for count data. Bernoulli, 20 (3), 1344-1371.

Junker, R.R., Griessenberger, F. and Trutschnig, W. (2021). Estimating scale-invariant directed dependence of bivariate distributions. Computational Statistics and Data Analysis, 153, 107058.

Examples

##Generate data drawn from the product copula and compute the empirical (checkerboard) copula
n <- 100
x <- runif(n, 0, 1)
y <- runif(n, 0, 1)
mass <- ECBC(x,y, resolution = 10)
plot_density(mass)
mass <- ECBC(x,y, resolution = n)
plot_density(mass)

## Compute empirical checkerboard copula of a sample with ties and plot density
n <- 100
x <- sample(runif(n, -1, 1), n, replace=TRUE)
y <- x^2 + rnorm(n, 0, 1)
mass <- ECBC(x,y)
plot_density(mass)