copBasic (version 2.0.1)

hoefCOP: The Hoeffding's Phi of a Copula or Lp Distances (Independence, Radial Asymmetry, or Reflection Symmetry Forms)

Description

Compute the measure of association known as Hoeffding's Phi $\Phi_\mathbf{C}$ of a copula from independence ($uv = \mathbf{\Pi}$; P) according to Cherunbini et al. (2004, p. 164) by $${\mathrm{\Phi }}_{\mathbf{C}}=3\sqrt{10\int {\int }_{{\mathcal{I}}^2}(\mathbf{C}(u,v)-uv{)}^2\mathrm{d}u\mathrm{d}v}\text{,}$$ and Nelsen (2006, p. 210) shows this as $${\mathrm{\Phi }}_{\mathbf{C}}=(90\int {\int }_{{\mathcal{I}}^2}|\mathbf{C}(u,v)-uv{|}^2\mathrm{d}u\mathrm{d}v{)}^{1/2}\text{,}$$ for which $\Phi^2_\mathbf{C}$ (the square of the quantity) is known as the dependence index. Gaißer{Gaisser} et al. (2010, eq. 1) have $\Phi^2_\mathbf{C}$ as Hoeffding's Phi-Square, and their definition, when square-rooted, matches Nelsen's listing.

A generalization (Nelsen, 2006) to $L_p$ distances from independence ($uv = \mathbf{\Pi}$; P) through the LpCOP function is $$L_p\equiv {\mathrm{\Phi }}_{\mathbf{C}}(p)=(k(p)\int {\int }_{{\mathcal{I}}^2}|\mathbf{C}(u,v)-uv{|}^p\mathrm{d}u\mathrm{d}v{)}^{1/p}\text{,}$$ for a $p: 1 \le p \le \infty$ and where $k(p)$ is a normalization constant such that $\Phi_\mathbf{C}(p) = 1$ when the copula $\mathbf{C}$ is $\mathbf{M}$ (see M) or $\mathbf{W}$ (see W). The $k(p)$ (bivariate definition only) for other powers is given (Nelsen, 2006, exer. 5.44, p. 213) in terms of the complete gamma function $\Gamma(t)$ by $$k(p)=\frac{\mathrm{\Gamma }(2p+3)}{2[\mathrm{\Gamma }(p+1){]}^2}\text{,}$$ which is implemented by the hoefCOP function. It is important to realize that the $L_p$ distances are symmetric nonparametric measures of dependence (Nelsen, 2006, p. 210). These are symmetric because distance from independence is used as evident by $uv$ in the above definitions.

Reflection/Radial and Permutation Asymmetry---Asymmetric forms similar to the above distances exist. Joe (2014, p. 65) shows two measures of bivariate reflection asymmetry or radial asymmetry (term favored in copBasic) as the distance between $\mathbf{C}(u,v)$ and the survival copula $\hat\mathbf{C}(u,v)$ (surCOP) measured by $$\text{Missing argument for mathbf}$$ or its $L_p^{(\mathrm{radsym})}$ counterpart $$\text{Missing argument for mathbf}$$ where $\hat\mathbf{C}(u,v) = u + v - 1 + \mathbf{C}(1-u, 1-v)$ and again $p: 1 \le p \le \infty$. Joe (2014) does not seem to discuss and normalization constants for these two radial asymmetry distances.

Joe (2014, p. 66) offers analogous measures of bivariate permutation asymmetry ($\mathbf{C}(u,v) \not= \mathbf{C}(v,u)$) defined as $$\text{Missing argument for mathbf}$$ or its $L_p^{(\mathrm{permsym})}$ counterpart $$\text{Missing argument for mathbf}$$ where $p: 1 \le p \le \infty$. Again, Joe (2014) does not seem to discuss and normalization constants for these two permutation symmetry distances. Joe (2014, p. 65) states that the simplest one-parameter bivariate copula families [and] most of the commonly used two-parameter bivariate copula families are permutation symmetric.

The asymmetrical $L_\infty$ and $L_p$ measures identified by Joe (2014, p. 66) are nonnegative with an upper bounds that depends on $p$. (The bound dependence on $p$ is likely(?) cause by the lack of normalization constant $k(p)$.) In an earlier paragraph, Joe indicates an upper bounds of 1/3 for both (likely?) concerning $L_\infty^{(\mathrm{radsym})}$ and $L_\infty^{(\mathrm{permsym})}$. The numerical integrations for $L_p^{(\mathrm{radsym})}$ and $L_p^{(\mathrm{permsym})}$ can readily return zeros and often inspection of the formula for the $\mathbf{C}$ itself would be sufficient to judge whether symmetry exists and hence the distances are uniquely zero.

Joe (2014, p. 66) completes the asymmetry discussion with three definitions of skewness of combinations of random variables $U$ and $V$: Two definitions are in uvlmoms (for $U + V - 1$ and $U - V$) and two are in joeskewCOP (for $V-U$ and $U+V-1$) where only the later function makes direct computations directly using formula for $\mathbf{C}$.

Usage

hoefCOP(cop=NULL,      para=NULL, p=2, as.sample=FALSE, sample.as.prob=TRUE,
                                       brute=FALSE, delta=0.002, ...)
LpCOP(cop=NULL,        para=NULL, p=2, brute=FALSE, delta=0.002, ...)
LpCOPradsym(cop=NULL,  para=NULL, p=2, brute=FALSE, delta=0.002, ...)
LpCOPpermsym(cop=NULL, para=NULL, p=2, brute=FALSE, delta=0.002, ...)

Arguments

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

p

The value for $p$ as described above with a default to 2 to match the discussion of Nelsen (2006) and Hoeffding's Phi of Cherubini et al. (2004). Do not confuse $p$ with $d$ described in Note;

as.sample

A logical controlling whether an optional Rdata.frame in para is used to compute the $\hat{\Phi}_\mathbf{C}$ (see Note). If set to -1, then the message concerning CPU effort will be surpressed;

sample.as.prob

When as.sample triggered, what are the units incoming in para? If they are probabilities, the default is applicable. If they are not, then the columns are re-ranked and divided simply by $1/n$---more sophisticated empirical c

brute

Should brute force be used instead of two nested integrate() functions in Rto perform the double integration;

delta

The $\mathrm{d}u$ and $\mathrm{d}v$ for the brute force (brute=TRUE) integration; and

...

Additional arguments to pass.

Value

• The value for $\Phi_\mathbf{C}(p)$ is returned.

encoding

utf8

concept

• Hoeffding Phi
• Hoeffding Phi-Square

References

Cherubini, U., Luciano, E., and Vecchiato, W., 2004, Copula methods in finance: Hoboken, NJ, Wiley, 293 p.

Gaißer{Gaisser}, S., Ruppert, M., and Schmid, F., 2010, A multivariate version of Hoeffding's Phi-Square: Journal of Multivariate Analysis, v. 101, no. 10, pp. 2571--2586.

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

See Also

blomCOP, footCOP, giniCOP, rhoCOP, tauCOP, wolfCOP, joeskewCOP, uvlmoms

Examples

# Example (ii) Gaisser et al. (2010, p. 2574)
Theta <- 0.66 # Phi^2 = Theta^2 ---> Phi == Theta as shown
hoefCOP(cop=convex2COP, para=c(alpha=Theta, cop1=M, cop2=P)) # 0.6599886

rhoCOP(cop=PSP) == hoefCOP(cop=PSP, p=1) # TRUE
LpCOP(cop=PLACKETTcop, para=1.6, p=2.6)  # 0.1445137 (Fractional p)

set.seed(938) # Phi(1.6; Plackett) = 0.1184489; L_1 = 0.1168737
UV <- simCOP(cop=PLACKETTcop, para=1.6, n=2000, ploton=FALSE, points=FALSE)
hoefCOP(cop=PLACKETTcop, para=1.6, p=200)  # Large p near internal limits
L_1 <- 4*max(abs(PLACKETTcop(UV$U, UV$V, para=1.6) - UV$U*UV$V)) # p is infty
# and finite n and arguably a sample-like statistic here, now on intuition try
# a more sample-like means
U <- runif(10000); V <- runif(10000)
L_2 <- 4*max(abs(EMPIRcop(U, V, para=UV) - U*V)) # 0.1410254 (not close enough)

para <- list(alpha=0.15, beta=0.90, kappa=0.06, gamma=0.96,
             cop1=GHcop, cop2=PLACKETTcop, para1=5.5, para2=0.07)
LpCOPradsym( cop=composite2COP, para=para) # 0.02071164
LpCOPpermsym(cop=composite2COP, para=para) # 0.01540297

"MOcop.formula" <- function(u,v, para=para, ...) {
   alpha <- para[1]; beta <- para[2]; return(min(v*u^(1-alpha), u*v^(1-beta)))
}
"MOcop" <- function(u,v, ...) { asCOP(u,v, f=MOcop.formula, ...) }
   LpCOPradsym( cop=MOcop, para=c(.8,.5)) # 0.0261843
   LpCOPpermsym(cop=MOcop, para=c(.8,.5)) # 0.0243912