BiCopEst.MO {MMDCopula}

Estimation of Marshall-Olkin copulas

Description

Estimation of Marshall-Olkin copulas

Usage

BiCopEst.MO(
  u1,
  u2,
  method,
  par.start = 0.5,
  kernel = "gaussian.Phi",
  gamma = 0.95,
  alpha = 1,
  niter = 100,
  C_eta = 1,
  ndrawings = 10,
  naveraging = 1
)

Arguments

u1

vector of observations of the first coordinate, in [0,1].

u2

vector of observations of the second coordinate, in [0,1].

method

a character giving the name of the estimation method, among:

• curve: \alpha is estimated by inversion of the probability measure of the diagonal \{(u,v): u = v\}

• itau: \alpha is estimated by inversion of Kendall's tau

• MMD: \alpha is estimated by MMD optimization

par.start

starting parameter of the gradient descent. (only used for method = "MMD")

kernel

the kernel used in the MMD distance (only used for method = "MMD") : it can be a function taking in parameter (u1, u2, v1, v2, gamma, alpha) or a name giving the kernel to use in the list:

• "gaussian": Gaussian kernel k(x,y) = \exp(-\|\frac{x-y}{\gamma}\|_2^2)

• "exp-l2": k(x,y) = \exp(-\|\frac{x-y}{\gamma}\|_2)

• "exp-l1": k(x,y) = \exp(-\|\frac{x-y}{\gamma}\|_1)

• "inv-l2": k(x,y) = 1/(1+\|\frac{x-y}{\gamma}\|_2)^\alpha

• "inv-l1": k(x,y) = 1/(1+\|\frac{x-y}{\gamma}\|_1)^\alpha

Each of these names can receive the suffix ".Phi", such as "gaussian.Phi" to indicates that the kernel k(x,y) is replaced by k(\Phi^{-1}(x) , \Phi^{-1}(y)) where \Phi^{-1} denotes the quantile function of the standard Normal distribution.

gamma

parameter \gamma to be used in the kernel. (only used for method = "MMD")

alpha

parameter \alpha to be used in the kernel, if any. (only used for method = "MMD")

niter

the stochastic gradient algorithm is composed of two phases: a first "burn-in" phase and a second "averaging" phase. If niter is of size 1, the same number of iterations is used for both phases of the stochastic gradient algorithm. If niter is of size 2, then niter[1] iterations are done for the burn-in phase and niter[2] for the averaging phase. (only used for method = "MMD")

C_eta

a multiplicative constant controlling for the size of the gradient descent step. The step size is then computed as C_eta / sqrt(i_iter) where i_iter is the index of the current iteration of the stochastic gradient algorithm. (only used for method = "MMD")

ndrawings

number of replicas of the stochastic estimate of the gradient drawn at each step. The gradient is computed using the average of these replicas. (only used for method = "MMD")

naveraging

number of full run of the stochastic gradient algorithm that are averaged at the end to give the final estimated parameter. (only used for method = "MMD")

Value

the estimated parameter (alpha) of the Marshall-Olkin copula.

References

Alquier, P., Chérief-Abdellatif, B.-E., Derumigny, A., and Fermanian, J.D. (2022). Estimation of copulas via Maximum Mean Discrepancy. Journal of the American Statistical Association, doi:10.1080/01621459.2021.2024836.

See Also

BiCopSim.MO for the estimation of Marshall-Olkin copulas. BiCopEstMMD for the estimation of other parametric copula families by MMD.

Examples

U <- BiCopSim.MO(n = 1000, alpha = 0.2)
estimatedPar <- BiCopEst.MO(u1 = U[,1], u2 = U[,2], method = "MMD", niter = 1, ndrawings = 1)

estimatedPar <- BiCopEst.MO(u1 = U[,1], u2 = U[,2], method = "MMD")