qad (version 0.2.0)

emp_c_copula: The empirical checkerboard copula

Description

The function emp_c_copula() computes the mass distribution of the empirical (checkerboard) copula, given a bivariate sample. emp_c_copula_eval() evaluates the the empirical (checkerboard) copula at given points. If smoothing = FALSE, the empirical copula is computed (if there are ties in the sample an adjusted empirical copula is computed), otherwise the empirical checkerboard copula - a smoothed version of the empirical copula - is computed. For more information of the calculations, see details.

Usage

emp_c_copula(X, smoothing = TRUE, resolution)
emp_c_copula_eval(X, u, smoothing = TRUE, resolution)

Arguments

X

a data frame with two columns containing the observations of the sample. Each row contains one observation.

smoothing

a logial indicating whether the checkerboard aggregation is computed (default = TRUE).

resolution

an integer indicating the resolution of the checkerboard aggregation, i.e. the number of vertical/horizontal strips of the checkerboard copula.

u

a data.frame with two columns containing the evaluation points. Each row consists of a x and y value.

Value

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

emp_c_copula_eval() returns a vector of evaluations of the empirical (checkerboard) copula.

Details

If the observations come from a distribution with continuous margins, i.e. there are no ties in the sample, the function emp_c_copula() gives the same result as the function C.n() in the copula package. 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.

The checkerboard aggregation is computed as usual (see references).

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.

Examples

# NOT RUN {
## Generate data X from the product copula and compute the empirical copula
n <- 1000
x <- runif(n, 0, 1)
y <- runif(n, 0, 1)
X <- data.frame(x,y)
#(Not Run)
# mass_product <- emp_c_copula(X, smoothing = TRUE, resolution = 50)
# eval_points <- data.frame(x = c(0.3,0.6), y = c(0.5,0.9))
# eval_points$emp_cop <- emp_c_copula_eval(X, eval_points, smoothing = TRUE, resolution = 50)
# eval_points$cop <- eval_points$x * eval_points$y

## Compute empirical checkerboard copula of a sample with ties and plot density
n <- 1000
x <- sample(runif(n, -1, 1), n, replace=TRUE)
y <- x^2 + rnorm(n, 0, 1)
X <- data.frame(x,y)
#(Not Run)
# mass <- emp_c_copula(X, smoothing = TRUE, resolution = 10)
# plot_density(mass)
# }