emp_c_copula | R Documentation |

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.

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

`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. |

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).

`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.

The calculation of the empirical copula with a high sample size (and resolution rate) can take time.

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.

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