# The Marshall--Olkin Copula The **Marshall--Olkin copula** (Dobrowolski and Kumar, 2014, eq. 2.6; Nelsen, 2006, p. 53), which is also known as the **Generalized copula**, having parameters $\alpha, \beta \in (0,1)$ is given by [REMOVE_ME]$$ \mathbf{C}_{(\alpha,\beta)}(u,v) = \mathrm{min}(u^{1-\alpha}v, uv^{1-\beta})\mbox{,\ } [REMOVE_ME_2]$$ and efficiently computed by compound equations for $u^\alpha \ge v^\beta$ [REMOVE_ME]$$ \mathbf{C}_{(\alpha,\beta)}(u,v) = \mathbf{MO}(u,v) = u^{1-\alpha}v\mbox{,} [REMOVE_ME_2]$$ and for $u^\alpha < v^\beta$ is [REMOVE_ME]$$ \mathbf{C}_{(\alpha,\beta)}(u,v) = \mathbf{MO}(u,v) = uv^{1-\beta}\mbox{.} [REMOVE_ME_2]$$ As $\alpha, \beta \rightarrow 1$, the copula limits to the **comonotonicity coupla** ($\mathbf{M}(u,v)$; `M`), and as $\alpha, \beta \rightarrow 0$, the copula limits to the **independence copula** ($\mathbf{\Pi}(u,v)$; `P`). The copula can be expected to have a visible singularity as the parameters increase and (or) a simulation size becomes large. The copula with $\alpha = \beta$ is **permutation symmetric** (`isCOP.permsym`, `breveCOP`) and is known as the **** copula. **Spearman Rho** (`rhoCOP`) of the copula is [REMOVE_ME]$$ \rho_\mathbf{C} = 3\alpha\beta / (2\alpha + 2\beta - \alpha\beta)\mbox{,\ } [REMOVE_ME_2]$$ and **Kendall tau** (`tauCOP`) is [REMOVE_ME]$$ \tau_\mathbf{C} = \alpha\beta / (\alpha + \beta - \alpha\beta)\mbox{.} [REMOVE_ME_2]$$ Parameter estimation using $\rho_\mathbf{C}$ and $\tau_\mathbf{C}$ is problematic because the two statistics are so correlated with each other and that either nonunique solutions or nonconverged solutions manifest with simultaneous numerical optimization. Also, application of **maximum likelihood** (`mleCOP`) can be problematic because of varying contributions of singularities that the copula can produce. The total contribution of singularity is given by $S(u\rightarrow1,v\rightarrow1) = \alpha\beta/ (\alpha + \beta - \alpha\beta)$, which Nelsen (2006, p. 165) remarks that it is interesting to note that Kendall tau is the singular component. Lastly, the **L-comoments** of copulas (`lcomCOP`) appear an attractive mechanism for parameter estimation not effected by singularities (refer to Examples ). The $\mathbf{MO}$ copula has a role in the three-parameter **Gumbel--Hougaard copula** copula (`GHcop`). utf8 ## Description The **Marshall--Olkin copula** (Dobrowolski and Kumar, 2014, eq. 2.6; Nelsen, 2006, p. 53), which is also known as the **Generalized copula**, having parameters $\alpha, \beta \in (0,1)$ is given by $$ \mathbf{C}_{(\alpha,\beta)}(u,v) = \mathrm{min}(u^{1-\alpha}v, uv^{1-\beta})\mbox{,\ } $$ and efficiently computed by compound equations for $u^\alpha \ge v^\beta$ $$ \mathbf{C}_{(\alpha,\beta)}(u,v) = \mathbf{MO}(u,v) = u^{1-\alpha}v\mbox{,} $$ and for $u^\alpha < v^\beta$ is $$ \mathbf{C}_{(\alpha,\beta)}(u,v) = \mathbf{MO}(u,v) = uv^{1-\beta}\mbox{.} $$ As $\alpha, \beta \rightarrow 1$, the copula limits to the **comonotonicity coupla** ($\mathbf{M}(u,v)$; `M`), and as $\alpha, \beta \rightarrow 0$, the copula limits to the **independence copula** ($\mathbf{\Pi}(u,v)$; `P`). The copula can be expected to have a visible singularity as the parameters increase and (or) a simulation size becomes large. The copula with $\alpha = \beta$ is **permutation symmetric** (`isCOP.permsym`, `breveCOP`) and is known as the **** copula. **Spearman Rho** (`rhoCOP`) of the copula is $$ \rho_\mathbf{C} = 3\alpha\beta / (2\alpha + 2\beta - \alpha\beta)\mbox{,\ } $$ and **Kendall tau** (`tauCOP`) is $$ \tau_\mathbf{C} = \alpha\beta / (\alpha + \beta - \alpha\beta)\mbox{.} $$ Parameter estimation using $\rho_\mathbf{C}$ and $\tau_\mathbf{C}$ is problematic because the two statistics are so correlated with each other and that either nonunique solutions or nonconverged solutions manifest with simultaneous numerical optimization. Also, application of **maximum likelihood** (`mleCOP`) can be problematic because of varying contributions of singularities that the copula can produce. The total contribution of singularity is given by $S(u\rightarrow1,v\rightarrow1) = \alpha\beta/ (\alpha + \beta - \alpha\beta)$, which Nelsen (2006, p. 165) remarks that it is interesting to note that Kendall tau is the singular component. Lastly, the **L-comoments** of copulas (`lcomCOP`) appear an attractive mechanism for parameter estimation not effected by singularities (refer to Examples ). The $\mathbf{MO}$ copula has a role in the three-parameter **Gumbel--Hougaard copula** copula (`GHcop`). ```r MOcop(u, v, para=NULL, lcomCOP=NULL, ...) ``` ## Arguments - `u`: Nonexceedance probability $u$ in the $X$ direction; - `v`: Nonexceedance probability $v$ in the $Y$ direction; - `para`: A vector (single element) of parameters---the $\alpha, \beta$ parameters of the copula; - `lcomCOP`: A vector containing $\rho_\mathbf{C}$ and the two L-coskews (refer to Examples ) from which the parameters are returned; and - `...`: Additional arguments to pass. ## Returns Value(s) for the copula are returned. Otherwise if either `lcomCOP` is given, then the $\alpha, \beta$ are computed by numerical optimization the the Spearman Rho and the two L-coskews (refer to package `lmomco` and function `lmomco::lcomoms2` for more details) (refer the demonstration of L-comoment parameter estimation in the Examples below). ## References Dobrowolski, E., and Kumar, P., 2014, Some properties of the Marshall--Olkin and generalized Cuadras-- familes of copula: Australian Journal of Mathematical Analysis and Applications: v. 11, no. 1, art. 2, pp. 1--13, accessed on August 10, 2025, at [https://ajmaa.org/searchroot/files/pdf/v11n1/v11i1p2.pdf](https://ajmaa.org/searchroot/files/pdf/v11n1/v11i1p2.pdf). Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p. ## Author(s) W.H. Asquith ## See Also `P`, `taildepCOP` ## Examples ```r ## Not run: # **** Demonstration of copula fitting by Method of L-comoments **** n <- 300 # large enough sample size to clearly show L-comoment viability UV <- simCOP(n, cop=MOcop, seed=1, para=c(0.3, 0.7), pch=16, col="seagreen") lcm <- lmomco::lcomoms2(UV) # sample L-comoments (mean off diag. T2 == Spearman Rho) sara <- MOcop(lcomCOP=c(mean(diag(lcm$T2[2:1,1:2])), lcm$T3[1,2], lcm$T3[2,1])) sara$para <- round(sara$para, digits=8) uv <- simCOP(n, cop=MOcop, seed=1, para=sara$para, col="blue", ploton=FALSE) message("MOcop : alpha=", sara$para[1], ", beta=", sara$para[2]) mara <- mleCOP(UV[,1], v=UV[,2], cop=MOcop, init.para=c(0.2,0.2), parafn=function(x) pnorm(x)) mara$para <- round(mara$para, digits=8) uv <- simCOP(n, cop=MOcop, seed=1, para=mara$para, col="red", ploton=FALSE) message("MOcop : alpha=", mara$para[1], ", beta=", mara$para[2]) # ## End(Not run) ## Not run: # This example developed as a demonstrator of the dissarray that the # Spearman Footrule measures as a nonunique relation to Spearman Rho. # The permutation asymmetry causes the relation between the two to # "paint" for positive association as shown in the graphic produced. cex <- c(1.3, 1.0, 0.7, 0.4) col <- c("#7D0112", "#67C7B1", "#1F28A2", "#CCC86C") pch <- c(1, 16, 1, 16) plot(c(-0.5, 1), c(0, 1), type="n", xlab="Spearman Footrule", ylab="") mtext(paste0("Absolute value of Spearman rho\n", "Actual sign is the same as for the Footrule"), side=2, line=2) mtext(paste0("Attainable Marshall-Olkin copula, MOcop() with permutation\n", "inclusion (breveCOP) and reflections (COP)"), line=1) for(r in 1:4) { # reflections, COP for(k in seq(-1, 1, by=0.1)) { # permutation asymmetry insertion, breveCOP for(a in seq(0, 1, by=0.11)) { # increment slightly different from beta for(b in seq(0, 1, by=0.09)) { # increment slightly different from alpha para <- list(cop=MOcop, para=c(a,b), breve=k, reflect=r) fotT <- footCOP(cop=breveCOP, para=para) rhoT <- rhoCOP(cop=breveCOP, para=para) points(fotT, abs(rhoT), lwd=0.8, pch=pch[r], cex=cex[r], col=col[r]) #UV <- simCOP(100, cop=breveCOP, para=para) } } } } # ## End(Not run) ```

MOcop function