hamse: hamse

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

hamse

Description

This function performs local bandwidth selection based on the amse (asymptotic mean squared error) for the beta kernel or Gaussian transformation kernel copula density estimator.

Usage

hamse(input, cop = NULL, pseudos = NULL, n, estimator, bw_method)

Arguments

input	The copula argument at which the optimal local bandwidth is to be computed.
cop	A fitted reference hac object, in case bw_method = 0 (default = NULL).
pseudos	The (estimated) copula observations from a q-dimensional random vector \mathbf{X} (n \times q matrix with observations in rows, variables in columns), in case bw_method = 1 (default = NULL).
n	The sample size.
estimator	Either "beta" or "trans" for the beta kernel or the Gaussian transformation kernel copula density estimator.
bw_method	A number in \{0,1\} specifying the method used for computing the bandwidth.

Details

When estimator = "beta", this function computes, at a certain input, a numerical approximation of the optimal local bandwidth (for the beta kernel copula density estimator) in terms of the amse (asymptotic mean squared error) given in equation (27) of De Keyser & Gijbels (2024). When estimator = "trans" (for the Gaussian transformation kernel copula density estimator), this optimal bandwidth is given at the end of Section 5.2 in De Keyser & Gijbels (2024).

Of course, these optimal bandwidths depend upon the true unknown copula. If bw_method = 0, then the given fitted (e.g., via MLE using mlehac) hac object (hierarchical Archimedean copula) cop is used as reference copula. If bw_method = 1, then a non-parametric (beta or Gaussian transformation) kernel copula density estimator based on the pseudos as pivot is used. This pivot is computed using the big O bandwidth (i.e., n^{-2/(q+4)} in case of the beta estimator, and n^{-1/(q+4)} for the transformation estimator, with q the total dimension).

Value

The optimal local bandwidth (in terms of amse).

References

De Keyser, S. & Gijbels, I. (2024). Hierarchical variable clustering via copula-based divergence measures between random vectors. International Journal of Approximate Reasoning 165:109090. doi: https://doi.org/10.1016/j.ijar.2023.109090.

See Also

betakernelestimator for the computation of the beta kernel copula density estimator,
transformationestimator for the computation of the Gaussian transformation kernel copula density estimator, phinp for fully non-parametric estimation of the \Phi-dependence between k random vectors.

Examples

q = 4
dim = c(2,2)

# Sample size
n = 1000

# Four dimensional hierarchical Gumbel copula
# with parameters (theta_0,theta_1,theta_2) = (2,3,4)
HAC = gethac(dim,c(2,3,4),type = 1)

# Sample
sample = suppressWarnings(HAC::rHAC(n,HAC))

# Copula pseudo-observations
pseudos = matrix(0,n,q)
for(j in 1:q){pseudos[,j] = (n/(n+1)) * ecdf(sample[,j])(sample[,j])}

# Maximum pseudo-likelihood estimator to be used
# as reference copula for bw_method = 0
est_cop = mlehac(sample,dim,1,c(2,3,4))

h_1 = hamse(rep(0.5,q),cop = est_cop,n = n,estimator = "beta",bw_method = 0)
h_2 = hamse(rep(0.5,q),cop = est_cop,n = n,estimator = "trans",bw_method = 0)
h_3 = hamse(rep(0.5,q),pseudos = pseudos,n = n,estimator = "beta",bw_method = 1)
h_4 = hamse(rep(0.5,q),pseudos = pseudos,n = n,estimator = "trans",bw_method = 1)

est_dens_1 = betakernelestimator(rep(0.5,q),h_1,pseudos)
est_dens_2 = transformationestimator(rep(0.5,q),h_2,pseudos)
est_dens_3 = betakernelestimator(rep(0.5,q),h_3,pseudos)
est_dens_4 = transformationestimator(rep(0.5,q),h_4,pseudos)

true = HAC::dHAC(c("X1" = 0.5, "X2" = 0.5, "X3" = 0.5, "X4" = 0.5), HAC)

VecDep documentation built on April 4, 2025, 5:14 a.m.