C.n {copula} | R Documentation |
Given pseudo-observations from a distribution with continuous margins and
copula C, the empirical copula is the empirical distribution function
of these pseudo-observations. It is thus a natural nonparametric estimator of
C. The function C.n()
computes the empirical copula or two
alternative smoothed versions of the latter: the empirical beta copula or the
empirical checkerboard copula; see Eqs. (2.1) and (4.1) in
Segers, Sibuya and Tsukahara (2016), and the references therein.
The function dCn()
approximates first-order partial derivatives
of the unknown copula using the empirical copula.
The function F.n()
computes the empirical distribution function
of a multivariate sample. Note that C.n(u, X, smoothing="none",
*)
simply calls F.n(u, pobs(X), *)
after checking u
.
C.n(u, X, smoothing=c("none", "beta", "checkerboard"), offset=0, method=c("C", "R")) dCn(u, U, j.ind=1:d, b=1/sqrt(nrow(U)), ...) F.n(x, X, offset=0, method=c("C", "R")) Cn(x, w) ## <-- deprecated! use C.n(w, x) instead!
u,w |
an (m, d)- |
x |
an (m, d)- |
U |
for |
X |
(and |
smoothing |
|
j.ind |
|
b |
|
offset |
used in scaling the result which is of the form
|
method |
|
... |
additional arguments passed to |
There are several asymptotically equivalent definitions of the
empirical copula. As mentioned above, the empirical copula
C.n(, smoothing = "none")
is simply defined as the empirical distribution function
computed from the pseudo-observations, that is,
(1/n) sum(I_{U_i<=u}, i=1, .., n),
where U_i, i=1,..,n,
denote the pseudo-observations (rows in U
) and n the
sample size. Internally, C.n(,smoothing = "none")
is just a wrapper for
F.n()
and is expected to be fed with the pseudo-observations.
The approximation for the jth partial derivative of the unknown copula C is implemented as, for example, in Rémillard and Scaillet (2009), and given by
hat(C.)[jn](u) = (C[n](u[1], .., u[j-1], min(u[j]+b, 1), u[j+1], .., u[d]) - C[n](u[1], .., u[j-1], max(u[j]-b, 0), u[j+1], .., u[d])) / (2b),
where b denotes the bandwidth and C[n] the empirical copula.
C.n()
returns the empirical copula at u
or a smoothed version of
the latter.
F.n()
returns the empirical distribution function of X
evaluated at
x
.
dCn()
returns a vector (length(j.ind)
is 1) or a matrix
(with number of columns equal to length(j.ind)
), containing
the approximated first-order partial derivatives of the unknown
copula at u
with respect to the arguments in j.ind
.
The first version of our empirical copula implementation, Cn()
,
had its two arguments reversed compared to C.n()
, and is
deprecated now. You must swap its arguments to transform to
new code.
The use of the two smoothed versions assumes implicitly no ties in the component samples of the data.
Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics, Annals of Statistics 4, 912–923.
Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: un test non paramétrique d'indépendance, Acad. Roy. Belg. Bull. Cl. Sci., 5th Ser. 65, 274–292.
Deheuvels, P. (1981). A non parametric test for independence, Publ. Inst. Statist. Univ. Paris 26, 29–50.
Rémillard, B. and Scaillet, O. (2009). Testing for equality between two copulas. Journal of Multivariate Analysis, 100(3), pages 377-386.
Segers, J., Sibuya, M. and Tsukahara, H. (2016). The Empirical Beta Copula. Journal of Multivariate Analysis, in press, arXiv:1607.04430.
pobs()
for computing pseudo-observations,
pCopula()
for evaluating a copula.
## Generate data X (from a meta-Gumbel model with N(0,1) margins) n <- 100 d <- 3 family <- "Gumbel" theta <- 2 cop <- onacopulaL(family, list(theta=theta, 1:d)) set.seed(1) X <- qnorm(rCopula(n, cop)) # meta-Gumbel data with N(0,1) margins ## Random points were to evaluate the empirical copula u <- matrix(runif(n*d), n, d) ec <- C.n(u, X) ## Compare the empirical copula with the true copula pc <- pCopula(u, copula=cop) mean(abs(pc - ec)) # ~= 0.012 -- increase n to decrease this error ## The same for the two smoothed versions beta <- C.n(u, X, smoothing = "beta") mean(abs(pc - beta)) check <- C.n(u, X, smoothing = "checkerboard") mean(abs(pc - check)) ## Compare the empirical copula with F.n(pobs()) U <- pobs(X) # pseudo-observations stopifnot(identical(ec, F.n(u, X=pobs(U)))) # even identical ## Compare the empirical copula based on U at U with the Kendall distribution ## Note: Theoretically, C(U) ~ K, so K(C_n(U, U=U)) should approximately be U(0,1) plot(pK(C.n(U, X), cop=cop@copula, d=d)) ## Compare the empirical copula and the true copula on the diagonal C.n.diag <- function(u) C.n(do.call(cbind, rep(list(u), d)), X=X) # diagonal of C_n C.diag <- function(u) pCopula(do.call(cbind, rep(list(u), d)), cop) # diagonal of C curve(C.n.diag, from=0, to=1, # empirical copula diagonal main=paste("True vs empirical diagonal of a", family, "copula"), xlab="u", ylab=quote("True C(u,..,u) and empirical"~C[n](u,..,u))) curve(C.diag, lty=2, add=TRUE) # add true copula diagonal legend("bottomright", lty=2:1, bty="n", inset=0.02, legend = expression(C, C[n])) ## Approximate partial derivatives w.r.t. the 2nd and 3rd component j.ind <- 2:3 # indices w.r.t. which the partial derivatives are computed ## Partial derivatives based on the empirical copula and the true copula der23 <- dCn(u, U=pobs(U), j.ind=j.ind) der23. <- copula:::dCdu(archmCopula(family, param=theta, dim=d), u=u)[,j.ind] ## Approximation error summary(as.vector(abs(der23-der23.))) ## For an example of using F.n(), see help(mvdc)% ./Mvdc.Rd