footCOP: The Spearman Footrule of a Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

The Spearman Footrule of a Copula

Description

Compute the measure of association known as the Spearman Footrule \psi_\mathbf{C} (Nelsen et al., 2001, p. 281), which is defined as

\psi_\mathbf{C} = \frac{3}{2}\mathcal{Q}(\mathbf{C},\mathbf{M}) - \frac{1}{2}\mbox{,}

where \mathbf{C}(u,v) is the copula, \mathbf{M}(u,v) is the Fréchet–Hoeffding upper bound (M), and \mathcal{Q}(a,b) is a concordance function (concordCOP) (Nelsen, 2006, p. 158). The \psi_\mathbf{C} in terms of a single integration pass on the copula is

\psi_\mathbf{C} = 1 - \int_{\mathcal{I}^2} |u-v|\,\mathrm{d}\mathbf{C}(u,v) = 6 \int_0^1 \mathbf{C}(u,u)\,\mathrm{d}u - 2\mbox{.}

Note, Nelsen et al. (2001) use \phi_\mathbf{C} but that symbol is taken in copBasic for the Hoeffding Phi (hoefCOP), and Spearman Footrule does not seem to appear in Nelsen (2006). From the definition, Spearman Footrule only depends on the primary diagnonal (alt. main diagonal, Genest et al., 2010) of the copula, \mathbf{C}(t,t) (diagCOP).

Usage

footCOP(cop=NULL, para=NULL, by.concordance=FALSE, as.sample=FALSE, ...)

Arguments

cop	A copula function;
para	Vector of parameters or other data structure, if needed, to pass to the copula;
by.concordance	Instead of using the single integral to compute \psi_\mathbf{C}, use the concordance function method implemented through concordCOP; and
as.sample	A logical controlling whether an optional R data.frame in para is used to compute the \hat\psi (see Note); and
...	Additional arguments to pass, which are dispatched to the copula function cop and possibly concordCOP, such as brute or delta used by that function.

Value

The value for \psi_\mathbf{C} is returned.

Note

Conceptually, the sample Spearman Footrule is a standardized sum of the absolute difference in the ranks (Genest et al., 2010). The sample \hat\psi is

\hat\psi = 1 - \frac{3}{n^2 - 1}\sum_{i=1}^n |R_i - S_i|\mbox{,}

where R_i and S_i are the respective ranks of X and Y and n is sample size. The sampling variance of \hat\psi under assumption of independence between X and Y is

\mathrm{var}(\hat\psi) = \frac{2n^2 + 7}{5(n+1)(n-1)^2}\mbox{.}

Genest et al. (2010, p. 938) say that prior literature shows that in small samples, Spearman Footrule is less variable than the well-known Spearman Rho (rhoCOP). For a copula having continuous partial derivatives, then as n \rightarrow \infty, the quantity (\hat\psi - \psi_\mathbf{C})\sqrt{n} \sim \mathrm{Normal}(0, \mathrm{var}(\gamma_\mathbf{C})). Genest et al. (2010) show variance of \hat\psi for the independence copula (\mathbf{C}(u,v) = \mathbf{\Pi}(u,v)) (P) as \mathrm{var}(\psi_\mathbf{C}) = 2/5. For comparison, the Gini Gamma for independence is larger at \mathrm{var}(\gamma_\mathbf{C}) = 2/3 (see giniCOP Note). Genest et al. (2010) also present additional material for estimation of the distribution \hat\psi variance for conditions of dependence based on copulas. In Genest et al. independence and two examples of dependence (p. 941), \mathrm{var}(\hat\gamma) > \mathrm{var}(\hat\psi), but those authors do not appear to remark on whether this inequality holds for all copula.

Author(s)

W.H. Asquith

References

Genest, C., Nešlehová, J., and Ghorbal, N.B., 2010, Spearman's footrule and Gini's gamma—A review with complements: Journal of Nonparametric Statistics, v. 22, no. 8, pp. 937–954, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10485250903499667")}.

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M., 2001, Distribution functions of copulas—A class of bivariate probability integral transforms: Statistics and Probability Letters, v. 54, no. 3, pp. 277–282, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0167-7152(01)00060-8")}.

See Also

blomCOP, giniCOP, hoefCOP, rhoCOP, tauCOP, wolfCOP

Examples

  footCOP(cop=PSP)                      # 0.3177662
# footCOP(cop=PSP, by.concordance=TRUE) # 0.3178025

## Not run: 
n <- 2000; UV <- simCOP(n=n, cop=GHcop, para=2.3, graphics=FALSE)
footCOP(para=UV, as.sample=TRUE)                  # 0.5594364 (sample version)
footCOP(cop=GHcop, para=2.3)                      # 0.5513380 (copula integration)
footCOP(cop=GHcop, para=2.3, by.concordance=TRUE) # 0.5513562 (concordance function)
# where the later issued warnings on the integration
## End(Not run)

## Not run: 
set.seed(1); nsim <- 1000
varFTunderIndpendence <- function(n) {
  (2*n^2 + 7) / (5*(n+1)*(n-1)^2) # Genest et al. (2010)
}
ns <- c(10, 15, 20, 25, 50, 75, 100)
plot(min(ns):max(ns), varFTunderIndpendence(10:max(ns)), type="l",
     xlab="Sample size", ylab="Variance of Sample Estimator", col="salmon4")
mtext("Sample Spearman Footrule Under Independence", col="salmon4")
for(n in ns) {
  sFT <- vector(length=nsim)
  for(i in seq_len(nsim)) {
    uv <- simCOP(n=n, cop=P, para=2, graphics=FALSE)
    sFT[i] <- footCOP(para=uv, as.sample=TRUE)
  }
  varFT <- varFTunderIndpendence(n)
  zz <- round(c(n, mean(sFT), var(sFT), varFT), digits=6)
  names(zz) <- c("n", "mean_sim", "var_sim", "var_Genest")
  print(zz)
  points(n, zz[3], cex=2, pch=21, col="salmon4", bg="salmon1")
} # results show proper implementation and Genest et al. (2010, sec. 3) 
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.