Maximum Approximate Bernstein Likelihood Estimate of Copula Density Function
mable.copula(
x,
M0 = 1,
M,
unif.mar = TRUE,
pseudo.obs = c("empirical", "mable"),
interval = NULL,
search = TRUE,
mar.deg = FALSE,
high.dim = FALSE,
controls = mable.ctrl(sig.level = 0.05),
progress = TRUE
)A list with components
m a vector of the selected optimal degrees by the method of change-point
p a vector of the mixture proportions \(p(j_1, \ldots, j_d)\), arranged in the
column-major order of \(j = (j_1, \ldots, j_d)\), \(0 \le j_i \le m_i, i = 1, \ldots, d\).
mloglik the maximum log-likelihood at an optimal degree m
pval the p-values of change-points for choosing the optimal degrees for the
marginal densities
M the vector (m1, m2, ..., md) at which the search of model degrees stopped.
If mar.deg=TRUE mi is the largest candidate degree when the search stoped for
the i-th marginal density
convergence An integer code. 0 indicates successful completion(the EM iteration is
convergent). 1 indicates that the iteration limit maxit had been reached in the EM iteration;
if unif.mar=FALSE, margin contains objects of the results of mable fit
to the marginal data
an n x d matrix or data.frame of multivariate sample of size n
from d-variate distribution with hyperrectangular specified by interval.
a nonnegative integer or a vector of d nonnegative integers specify
starting candidate degrees for searching optimal degrees.
a positive integer or a vector of d positive integers specify
the maximum candidate or the given model degrees for the joint density.
logical, whether all the marginals distributions are uniform or not.
If not the pseudo observations will be created using empirical or mable
marginal distributions.
"empirical": use empirical distribution to create pseudo,
observations, or "mable": use mable of marginal cdfs to create pseudo observations
a vector of two endpoints or a 2 x d matrix, each column containing
the endpoints of support/truncation interval for each marginal density.
If missing, the i-th column is assigned as extendrange(x[,i]).
If unif.mar=TRUE, then it is \([0,1]^d\).
logical, whether to search optimal degrees between M0 and M
or not but use M as the given model degrees for the joint density.
logical, if TRUE (default), the optimal degrees are selected based on marginal data, otherwise, the optimal degrees are chosen by the method of change-point. See details.
logical, data are high dimensional/large sample or not if TRUE, run a slower version procedure which requires less memory
Object of class mable.ctrl() specifying iteration limit
and the convergence criterion eps. Default is mable.ctrl. See Details.
if TRUE a text progressbar is displayed
Zhong Guan <zguan@iu.edu>
A \(d\)-variate copula density \(c(u)\) on \([0, 1]^d\) can be approximated
by a mixture of \(d\)-variate beta densities on \([0, 1]^d\),
\(\beta_{mj}(x) = \prod_{i=1}^d\beta_{m_i,j_i}(u_i)\),
with proportion \(p(j_1, \ldots, j_d)\), \(0 \le j_i \le m_i, i = 1, \ldots, d\),
which satisfy the uniform marginal constraints, the copula (density) has
uniform marginal cdf (pdf). If search=TRUE and mar.deg=TRUE, then the
optimal degrees are \((\tilde m_1,\ldots,\tilde m_d)\), where \(\tilde m_i\) is
chosen based on marginal data of \(u_i\), $\(i=1,\ldots,d\). If search=TRUE
and mar.deg=FALSE, then the optimal degrees \((\hat m_1,\ldots,\hat m_d)\)
are chosen using a change-point method based on the joint data.
For large data and high dimensional density, the search for the model degrees might be time-consuming. Thus patience is needed.
Wang, T. and Guan, Z. (2019). Bernstein polynomial model for nonparametric multivariate density. Statistics 53(2), 321–338. Guan, Z., Nonparametric Maximum Likelihood Estimation of Copula
mable, mable.mvar
## Simulated bivariate data from Gaussian copula
# \donttest{
set.seed(1)
rho<-0.4; n<-1000
x<-rnorm(n)
u<-pnorm(cbind(rnorm(n, mean=rho*x, sd=sqrt(1-rho^2)),x))
res<- mable.copula(u, M = c(3,3), search =FALSE, mar.deg=FALSE, progress=FALSE)
plot(res, which="density")
# }
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