copula (version 0.999-15)

K: Kendall Distribution Function for Archimedean Copulas

Description

The distribution function of the Kendall distribution of an Archimedean copula is defined as $$K(u) = P(C(U_1,U_2,\dots,U_d) \le u),$$ where $u in [0,1]$, and the $d$-dimensional $(U[1],U[2],\dots,U[d])$ is distributed according to the copula $C$. Note that the random variable $C(U[1],U[2], \dots, U[d])$ is known as “probability integral transform”. Its distribution function $K$ is equal to the identity if $d = 1$, but is non-trivial for $d >= 2$.

Kn() computes the empirical Kendall distribution function, pK(), qK(), dK(), and rK() provide the distribution function, quantile function, density, and random number generator, respectively, for the Kendall distribution of an Archimedean copula.

Note that R function pK() provides the $K()$ function.

Usage

Kn(u, x)
pK(u, copula, d, n.MC=0, log.p=FALSE)
qK(u, copula, d, n.MC=0, log.p=FALSE, method=c("default", "simple", "sort", "discrete", "monoH.FC"), u.grid, ...)
dK(u, copula, d, n.MC=0, log.p=FALSE)
rK(n, copula, d)

Arguments

u

evaluation point(s) (have to be in $[0,1]$).

x

data (in the $d$-dimensional space) based on which the Kendall distribution function is estimated.

copula

acopula with specified parameter, or (currently for rK only) a outer_nacopula.

d

dimension (not used when copula is an outer_nacopula).

n.MC

integer, if positive, a Monte Carlo approach is applied with sample size equal to n.MC to evaluate the generator derivatives involved; otherwise (n.MC=0) the exact formula is used based on the generator derivatives as found by Hofert et al. (2012).

log.p

logical; if TRUE, probabilities $p$ are given as $log(p)$.

method

string for the method to compute the quantile function of $K$. Currently, one of

u.grid

(for method="discrete":) the grid on which $K$ is evaluated, a numeric vector.

...

additional arguments passed to uniroot (for method="default", method="simple", method="sort", and method="monoH.FC") or findInterval (for method="discrete").

n

sample size for rK.

Value

$K_n$, the Kendall distribution function $K(.)$, its quantile function, density, and random number generator.

Details

For a completely monotone Archimedean generator $psi$, $$K(u)=\sum_{k=0}^{d-1} \frac{\psi^{(k)}(\psi^{-1}(u))}{k!} (-\psi^{-1}(u))^k,\ u\in[0,1];$$ see Barbe et al. (1996). The corresponding density is $$\frac{(-1)^d\psi^{(d)}(\psi^{-1}(u))}{(d-1)!} (-(\psi^{-1})'(u))(\psi^{-1}(u))^{d-1}$$

The empirical Kendall distribution function is computed as in Genest, G. Nešlehová, Ziegel (2011).

References

Barbe, P., Genest, C., Ghoudi, K., and Rémillard, B. (1996), On Kendall's Process, Journal of Multivariate Analysis 58, 197--229.

Hofert, M., Mächler, M., and McNeil, A. J. (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis 110, 133--150.

Genest, C., G. Nešlehová, J., and Ziegel, J. (2011). Inference in multivariate Archimedean copula models. TEST 20, 223--256.

See Also

htrafo or emde (where K is used); splinefun(*, "monoHC") for that method.

Examples

tau <- 0.5
(theta <- copGumbel@iTau(tau)) # 2
d <- 20
(cop <- onacopulaL("Gumbel", list(theta,1:d)))

## Basic check empirical Kendall distribution function
set.seed(271)
n <- 1000
U <- rCopula(n, copula = cop)
X <- qnorm(U)
K.sample <- pCopula(U, copula = cop)
u <- seq(0, 1, length.out=256)
edfK <- ecdf(K.sample)
plot(u, edfK(u), type="l", ylim=c(0,1),
     xlab=expression(italic(u)), ylab=expression(K[n](italic(u)))) # simulated
K.n <- Kn(u, x=X)
lines(u, K.n, col="royalblue3") # Kn
## Difference at 0
edfK(0) # edf of K at 0
K.n[1] # K_n(0); this is > 0 since K.n is the edf of a discrete distribution
## => therefore, Kn(K.sample, x=X) is not uniform
plot(Kn(K.sample, x=X), ylim=c(0,1))
## Note: Kn(0) -> 0 for n -> Inf

## Compute Kendall distribution function
u <- seq(0,1, length.out = 255)
Ku    <- pK(u, copula = cop@copula, d = d) # exact
Ku.MC <- pK(u, copula = cop@copula, d = d, n.MC = 1000) # via Monte Carlo

## Build sample from K
set.seed(1)
n <- 200
W <- rK(n, copula = cop)

## Plot empirical distribution function based on W
## and the corresponding theoretical Kendall distribution function
## (exact and via Monte Carlo)
plot(ecdf(W), col="blue", xlim=c(0,1), verticals=TRUE,
     main = expression("Empirical"~ F[n]( C(U) ) ~
                       "and its Kendall distribution"~ K(u)),
     do.points=FALSE, asp=1)
abline(0,1, lty=2); abline(h=0:1, v=0:1, lty=3, col="gray")
lines(u, Ku.MC, col="red") # not quite monotone
lines(u, Ku, col="black")  # strictly  monotone:
stopifnot(diff(Ku) >= 0)
legend(.25, .75, expression(F[n], K[MC](u), K(u)),
       col=c("blue" , "red", "black"), lty=1, lwd = 1.5, bty="n")

## Testing qK
uexpr <- quote( 0:63/63 );  u <- eval(uexpr)
d <- 10
cop <- onacopulaL("Gumbel", list(theta = 2, 1:d))
system.time(qK0 <- qK(u, copula = cop@copula, d = d)) # "default" - fast

system.time(qK1 <- qK(u, copula = cop@copula, d = d, method = "simple"))
system.time(qK2 <- qK(u, copula = cop@copula, d = d, method = "sort"))
system.time(qK3 <- qK(u, copula = cop@copula, d = d, method = "discrete",
                      u.grid = 0:1e4/1e4))
system.time(qK4 <- qK(u, copula = cop@copula, d = d, method = "monoH.FC",
                      u.grid = 0:5e2/5e2))
system.time(qK5 <- qK(u, copula = cop@copula, d = d, method = "monoH.FC",
                      u.grid = 0:5e3/5e3))
system.time(qK6 <- qK(u, copula = cop@copula, d = d, method = "monoH.FC",
                      u.grid = c(0, (1:5e3/5e3)^2)))

## Visually they all coincide :
cols <- adjustcolor(c("gray50", "gray80", "light blue",
                      "royal blue", "purple3", "purple4", "purple"), 0.6)
matplot(u, cbind(qK0, qK1, qK2, qK3, qK4, qK5, qK6), type = "l", lwd = 2*7:1, lty = 1:7, col = cols,
        xlab = bquote(u == .(uexpr)), ylab = quote({K^{-1}}(u)),
        main = "qK(u, method = *)")
legend("topleft", col = cols, lwd = 2*7:1, lty = 1:7, bty = "n", inset = .03,
       legend= paste0("method= ",
             sQuote(c("default", "simple", "sort",
                      "discrete(1e4)", "monoH.FC(500)", "monoH.FC(5e3)", "monoH.FC(*^2)"))))

## See they *are* inverses  (but only approximately!):
all.equal(u, pK(qK0, cop@copula, d=d)) # "default"	 0.03  worst
all.equal(u, pK(qK1, cop@copula, d=d)) # "simple"	 0.0011 - best
all.equal(u, pK(qK2, cop@copula, d=d)) # "sort"		 0.0013 (close)
all.equal(u, pK(qK3, cop@copula, d=d)) # "discrete"	 0.0026
all.equal(u, pK(qK4, cop@copula, d=d)) # "monoH.FC(500)" 0.0095
all.equal(u, pK(qK5, cop@copula, d=d)) # "monoH.FC(5e3)" 0.001148 - (best)
all.equal(u, pK(qK6, cop@copula, d=d)) # "monoH.FC(*^2)" 0.001111 - best

## and ensure the differences are not too large
stopifnot(
 all.equal(qK0, qK1, tol = 1e-2) # !
 ,
 all.equal(qK1, qK2, tol = 1e-4)
 ,
 all.equal(qK2, qK3, tol = 1e-3)
 ,
 all.equal(qK3, qK4, tol = 1e-3)
 ,
 all.equal(qK4, qK0, tol = 1e-2) # !
)


stopifnot(all.equal(u, pK(qK0, cop@copula, d=d), tol = 0.04))