K {copula} | R Documentation

## 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], …, U[d]) <= u),

where u in [0,1], and the d-dimensional (U[1],U[2],…,U[d]) is distributed according to the copula C. Note that the random variable C(U[1],U[2], …, 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

"default"

  The default method. Currently chooses `method="monoH.FC"` with `u.grid = 0:128/128`. This is fast but not too accurate (see example).

"simple"

  Straightforward root finding based on `uniroot`.

"sort"

  Root finding based on `uniroot` but first sorting `u`.

"discrete"

  First, K is evaluated at the given grid points `u.grid` (which should contain 0 and 1). Based on these probabilities, quantiles are computed with `findInterval`.

"monoH.FC"

  First, K is evaluated at the given grid points `u.grid`. A monotone spline is then used to approximate K. Based on this approximation, quantiles are computed with `uniroot`.

`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`.

### Details

For a completely monotone Archimedean generator psi,

K(u)=sum(k=0,...,d-1) psi^{(k)}(psi^{-1}(u))/k! (-psi^{-1}(u))^k, u in [0,1];

see Barbe et al. (1996). The corresponding density is

(-1)^d psi^{(d)}(psi^{-1}(u))(-(psi^{-1})'(u))(psi^{-1}(u))^{d-1}/(d-1)!

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

### Value

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

### Note

Currently, `qK()` is not very accurate

### 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))

[Package copula version 0.999-15 Index]