copBasic (version 2.2.7)

copBasic-package: Basic Theoretical Copula, Empirical Copula, and Various Utility Functions

Description

The copBasic package is oriented around bivariate copula theory and mathematical operations closely follow the recommended texts of Nelsen (2006) and Joe (2014) as well as select other references. Another recommended text is Salvadori et al. (2007) and Hofert et al. (2018) and are cited herein, but about half of that excellent book concerns univariate applications. The primal objective of copBasic is to provide a study of numerous results shown by authoritative texts on copulas. It is intended that the package will help other copula students in self study, potential course work, and applied circumstances.

Notes on copulas that are supported. The author has focused on pedagogical aspects of copulas, and this package is a diary of sorts beginning in fall 2008. Originally, the author did not implement many copulas in the copBasic in order to deliberately avoid redundancy to that support such as it exists on the R CRAN. Though as time has progressed, other copulas have been added occasionally based on needs of the user community, need to show some specific concept in the general theory, or test algorithms. For example, the Clayton copula (CLcop) was a late arriving addition the package (c.2017), which was added to assist a specific user; the Frank copula (FRcop was added in December 2024 because of an inquiry on using copBasic for vine copula (see vine example under FRcop).

The language and vocabulary of copulas are formidable even within the realm of bivariate or bicopula as design basis for the package. Often “vocabulary” words are often emphasized in italics, which is used extensively and usually near the opening of function-by-function documentation to identify vocabulary words, such as survival copula (see surCOP). This syntax tries to mimic and accentuate the terminology in Nelsen (2006), Joe (2014), and generally most other texts. Italics are used to draw connections between concepts.

In conjunction with the summary of functions in copBasic-package, the extensive cross referencing to functions and expansive keyword indexing should be beneficial. The author had no experience with copulas prior to a chance happening upon Nelsen (2006) in c.2008. The copBasic package is a personal tour de force in self-guided learning. Over time, this package and user's manual have been helpful to others.

Helpful Navigation of Copulas Implemented in the copBasic Package

Some entry points to the copulas implemented are listed in the Table of Copulas:

Name	Symbol	Function	Concept
Lower-bounds copula	$\mathbf{W}(u,v)$	W	copula
Independence copula	$\mathbf{\Pi }(u,v)$	P	copula
Upper-bounds copula	$\mathbf{M}(u,v)$	M	copula
Fréchet Family copula	$\mathbf{FF}(u,v)$	FRECHETcop	copula
Ali--Mikhail--Haq copula	$\mathbf{AMH}(u,v)$	AMHcop	copula
Clayton copula	$\mathbf{CL}(u,v)$	CLcop	copula
Copula of uniform circle	$\mathbf{CIRC}(u,v)$	CIRCcop	copula
Farlie--Gumbel--Morgenstern (generalized)	$\mathbf{FGM}(u,v)$	FGMcop	copula
Frank	$\mathbf{FR}(u,v)$	FRcop	copula
Galambos copula	$\mathbf{GL}(u,v)$	GLcop	copula
Gumbel--Hougaard copula	$\mathbf{GH}(u,v)$	GHcop	copula
Hüsler--Reiss copula	$\mathbf{HR}(u,v)$	HRcop	copula
Joe B5 (the “Joe”) copula	$\mathbf{B}\mathbf{5}(u,v)$	JOcopB5	copula
Nelsen eq.4-2-12 copula	$\mathbf{N}\mathbf{4212}\mathbf{cop}(u,v)$	N4212cop	copula
Ordinal Sums by Copula	${\mathbf{C}}_{\mathcal{J}}(u,v)$	ORDSUMcop	copula
Pareto copula	$\mathbf{PA}(u,v)$	PLcop	copula
Plackett copula	$\mathbf{PL}(u,v)$	PLcop	copula
PSP copula	$\mathbf{PSP}(u,v)$	PSP	copula
Raftery copula	$\mathbf{RF}(u,v)$	RFcop	copula
Rayleigh copula	$\mathbf{RAY}(u,v)$	RAYcop	copula
g-EV copula (Gaussian extreme value)	$\mathbf{gEV}(u,v)$	gEVcop	copula
t-EV copula (t-distribution extreme value)	$\mathbf{tEV}(u,v)$	tEVcop	copula

A few comments on notation herein are needed. A bold math typeface is used to represent a copula function or family such as $\mathbf{\Pi }$ (see P) for the independence copula. The syntax $\mathcal{R}\times \mathcal{R}\equiv {\mathcal{R}}^2$ denotes the orthogonal domain of two real numbers, and $[0,1]\times [0,1]$ $\equiv$ $\mathcal{I}\times \mathcal{I}\equiv {\mathcal{I}}^2$ denotes the orthogonal domain on the unit square of probabilities. Limits of integration $[0,1]$ or $[0,1{]}^2$ involving copulas are thus shown as $\mathcal{I}$ and ${\mathcal{I}}^2$, respectively.

Random variables $X$ and $Y$ respectively denote the horizontal and vertical directions in ${\mathcal{R}}^2$. Their probabilistic counterparts are uniformly distributed random variables on $[0,1]$, are respectively denoted as $U$ and $V$, and necessarily also are the respective directions in ${\mathcal{I}}^2$ ($U$ denotes the horizontal, $V$ denotes the vertical). Often realizations of these random variables are respectively $x$ and $y$ for $X$ and $Y$ and $u$ and $v$ for $U$ and $V$.

There is an obvious difference between nonexceedance probability $F$ and its complement, which is exceedance probability defined as $1-F$. Both $u$ and $v$ herein are in nonexceedance probability. Arguments to many functions herein are u $=u$ and v $=v$ and are almost exclusively nonexceedance but there are instances for which the probability arguments are u $=1-u=u^{\prime }$ and v $=1-v=v^{\prime }$.

Several of the functions listed above are measures of “bivariate association.” Two of the measures (Kendall Tau, tauCOP; Spearman Rho, rhoCOP) are widely known. R provides native support for their sample estimation of course, but each function can be used to call the cor() function in R for parallelism to the other measures of this package. The other measures (Blomqvist Beta, Gini Gamma, Hoeffding Phi, Schweizer--Wolff Sigma, Spearman Footrule) support sample estimation by specially formed calls to their respective functions: blomCOP, giniCOP, hoefCOP, wolfCOP, and footCOP. Gini Gamma (giniCOP) documentation (also joeskewCOP) shows extensive use of theoretical and sample computations for these and other functions.

Helpful Navigation of the copBasic Package

Some other entry points into the package are listed in the following table:

Name	Symbol	Function	Concept
Copula	$\mathbf{C}(u,v)$	COP	copula theory
$\cdots$	$\mathbf{C}(u,v)$	COP(..., reflect=*)	reflection
$\cdots$	$\mathbf{C}(u,v)$	COP(..., reflect=*)	rotation
Survival copula	$\hat{\mathbf{C}}(u^{\prime },v^{\prime })$	surCOP	copula theory
Joint survival function	$\bar{\mathbf{C}}(u,v)$	surfuncCOP	copula theory
Co-copula	${\mathbf{C}}^{\star }(u^{\prime },v^{\prime })$	coCOP	copula theory
Dual of a copula	$\tilde{\mathbf{C}}(u,v)$	duCOP	copula theory
Primary copula diagonal	$\delta (t)$	diagCOP	copula theory
Secondary copula diagonal	${\delta }^{\star }(t)$	diagCOP	copula theory
Inverse copula diagonal	${\delta }^{(-1)}(f)$	diagCOPatf	copula theory
Joint probability	$--$	jointCOP	copula theory
Bivariate L-moments	${\delta }_{k;\mathbf{C}}^{[\dots ]}$	bilmoms and lcomCOP	bivariate moments
Bivariate L-comoments	${\tau }_{k;\mathbf{C}}^{[\dots ]}$	bilmoms and lcomCOP	bivariate moments
Blomqvist Beta	${\beta }_{\mathbf{C}}$	blomCOP	bivariate association
Gini Gamma	${\gamma }_{\mathbf{C}}$	giniCOP	bivariate association
Hoeffding Phi	${\mathrm{\Phi }}_{\mathbf{C}}$	hoefCOP	bivariate association
Nu-Skew	${\nu }_{\mathbf{C}}$	nuskewCOP	bivariate moments
Nu-Star (skew)	${\nu }_{\mathbf{C}}^{\star }$	nustarCOP	bivariate moments
Lp distance to independence	${\mathrm{\Phi }}_{\mathbf{C}}\to L_p$	LpCOP	bivariate association
Permutation-Mu	${\mu }_{\mathrm{\infty }\mathbf{C}}^{\mathrm{permsym}}$	LzCOPpermsym	permutation asymmetry
Kendall Tau	${\tau }_{\mathbf{C}}$	tauCOP	bivariate association
Kendall Measure	$K_{\mathbf{C}}(z)$	kmeasCOP	copula theory
Kendall Function	$F_K(z)$	kfuncCOP	copula theory
Inverse Kendall Function	$F_K^{(-1)}(z)$	kfuncCOPinv	copula theory
An L-moment of $F_K(z)$	${\lambda }_r(F_K)$	kfuncCOPlmom	L-moment theory
L-moments of $F_K(z)$	${\lambda }_r(F_K)$	kfuncCOPlmoms	L-moment theory
Semi-correlations (negatives)	${\rho }_N^{-}(a)$	semicorCOP	bivariate tail association
Semi-correlations (positives)	${\rho }_N^{+}(a)$	semicorCOP	bivariate tail association
Spearman Footrule	${\psi }_{\mathbf{C}}$	footCOP	bivariate association
Spearman Rho	${\rho }_{\mathbf{C}}$	rhoCOP	bivariate association
Schweizer--Wolff Sigma	${\sigma }_{\mathbf{C}}$	wolfCOP	bivariate association
Density of a copula	$c(u,v)$	densityCOP	copula density
Density visualization	$--$	densityCOPplot	copula density
Empirical copula	${\mathbf{C}}_n(u,v)$	EMPIRcop	copula
Empirical simulation	$--$	EMPIRsim	copula simulation
Empirical simulation	$--$	EMPIRsimv	copula simulation
Empirical copulatic surface	$--$	EMPIRgrid	copulatic surface
Parametric copulatic surface	$--$	gridCOP	copulatic surface
Parametric simulation	$--$	simCOP or rCOP	copula simulation
Parametric simulation	$--$	simCOPmicro	copula simulation
Maximum likelihood	$\mathcal{L}({\mathrm{\Theta }}_d)$	mleCOP	copula fitting
Akaike information criterion	${\mathrm{AIC}}_{\mathbf{C}}$	aicCOP	goodness-of-fit
Bayesian information criterion	${\mathrm{BIC}}_{\mathbf{C}}$	bicCOP	goodness-of-fit
Root mean square error	${\mathrm{RMSE}}_{\mathbf{C}}$	rmseCOP	goodness-of-fit
Another goodness-of-fit	$T_n$	statTn	goodness-of-fit

Concerning goodness-of-fit and although not quite the same as copula properties (such as “correlation”) per se as the coefficients aforementioned in the prior paragraph, three goodness-of-fit metrics of a copula compared to the empirical copula, which are all based the mean square error (MSE), are aicCOP, bicCOP, and rmseCOP. This triad of functions is useful for making decisions on whether a copula is more favorable than another to a given dataset. However, because they are genetically related by using MSE and if these are used for copula fitting by minimization, the fits will be identical. A statement of “not quite the same” is made because the previously described copula properties are generally defined as types of deviations from other copulas (such as P). Another goodness-of-fit statistic is statTn, which is based on magnitude summation of fitted copula difference from the empirical copula. These four (aicCOP, bicCOP, rmseCOP, and statTn) collectively are relative simple and readily understood measures. These bulk sample statistics are useful, but generally thought to not capture the nuances of tail behavior (semicorCOP and taildepCOP might be useful).

Bivariate skewness measures are supported in the functions joeskewCOP (nuskewCOP and nustarCOP) and uvlmoms (uvskew). Extensive discussion and example computations of bivariate skewness are provided in the joeskewCOP documentation. Lastly, so-called bivariate L-moments and bivariate L-comoments of a copula are directly computable in bilmoms (though that function using Monte Carlo integration is deprecated) and lcomCOP (direct numerical integration). The lcomCOP function is the theoretical counterpart to the sample L-comoments provided in the lmomco package.

Bivariate random simulation methods by several functions are identified in the previous table. The copBasic package explicitly uses only conditional simulation also known as the conditional distribution method for random variate generation following Nelsen (2006, pp. 40--41) (see also simCOPmicro, simCOP). The numerical derivatives (derCOP and derCOP2) and their inversions (derCOPinv and derCOPinv2) represent the foundation of the conditional simulation. There are other methods in the literature and available in other R packages, and a comparison of some methods is made in the Examples section of the Gumbel--Hougaard copula (GHcop).

Several functions in copBasic make the distinction between $V$ with respect to (wrt) $U$ and $U$ wrt $V$, and a guide for the nomenclature involving wrt distinctions is listed in the following table:

Name	Symbol	Function	Concept
Copula inversion	$V$ wrt $U$	COPinv	copula operator
Copula inversion	$U$ wrt $V$	COPinv2	copula operator
Copula derivative	$\delta \mathbf{C}/\delta u$	derCOP	copula operator
Copula derivative	$\delta \mathbf{C}/\delta v$	derCOP2	copula operator
Copula derivative inversion	$V$ wrt $U$	derCOPinv	copula operator
Copula derivative inversion	$U$ wrt $V$	derCOPinv2	copula operator
Joint curves	$t\mapsto \mathbf{C}(u=U,v)$	joint.curvesCOP	copula theory
Joint curves	$t\mapsto \mathbf{C}(u,v=V)$	joint.curvesCOP2	copula theory
Level curves	$t\mapsto \mathbf{C}(u=U,v)$	level.curvesCOP	copula theory
Level curves	$t\mapsto \mathbf{C}(u,v=V)$	level.curvesCOP2	copula theory
Level set	$V$ wrt $U$	level.setCOP	copula theory
Level set	$U$ wrt $V$	level.setCOP2	copula theory
Median regression	$V$ wrt $U$	med.regressCOP	copula theory
Median regression	$U$ wrt $V$	med.regressCOP2	copula theory
Quantile regression	$V$ wrt $U$	qua.regressCOP	copula theory
Quantile regression	$U$ wrt $V$	qua.regressCOP2	copula theory
Copula section	$t\mapsto \mathbf{C}(t,a)$	sectionCOP	copula theory
Copula section	$t\mapsto \mathbf{C}(a,t)$	sectionCOP	copula theory

The previous two tables do not include all of the myriad of special functions to support similar operations on empirical copulas. All empirical copula operators and utilities are prepended with EMPIR in the function name. An additional note concerning package nomenclature is that an appended “2” to a function name indicates $U$ wrt $V$ (e.g. EMPIRgridderinv2 for an inversion of the partial derivatives $\delta \mathbf{C}/\delta v$ across the grid of the empirical copula).

Some additional functions to compute often salient features or characteristics of copulas or bivariate data, including functions for bivariate inference or goodness-of-fit, are listed in the following table:

Name	Symbol	Function	Concept
Left-tail decreasing	$V$ wrt $U$	isCOP.LTD	bivariate association
Left-tail decreasing	$U$ wrt $V$	isCOP.LTD	bivariate association
Right-tail increasing	$V$ wrt $U$	isCOP.RTI	bivariate association
Right-tail increasing	$U$ wrt $V$	isCOP.RTI	bivariate association
Pseudo-polar representation	$(\hat{S},\hat{W})$	psepolar	extremal dependency
Tail concentration function	$q_{\mathbf{C}}(t)$	tailconCOP	bivariate tail association
Tail (lower) dependency	${\lambda }_{\mathbf{C}}^L$	taildepCOP	bivariate tail association
Tail (upper) dependency	${\lambda }_{\mathbf{C}}^U$	taildepCOP	bivariate tail association
Tail (lower) order	${\kappa }_{\mathbf{C}}^L$	tailordCOP	bivariate tail association
Tail (upper) order	${\kappa }_{\mathbf{C}}^U$	tailordCOP	bivariate tail association
Neg'ly quadrant dependency	NQD	isCOP.PQD	bivariate association
Pos'ly quadrant dependency	PQD	isCOP.PQD	bivariate association
Permutation symmetry	$\mathrm{permsym}$	isCOP.permsym	copula symmetry
Radial symmetry	$\mathrm{radsym}$	isCOP.radsym	copula symmetry
Skewness (Joe, 2014)	$\eta (p;\psi )$	uvskew	bivariate skewness
Kullback--Leibler Divergence	$\mathrm{KL}(f\mid g)$	kullCOP	bivariate inference
KL sample size	$n_{fg}$	kullCOP	bivariate inference
The Vuong Procedure	$--$	vuongCOP	bivariate inference
Spectral measure	$H(w)$	spectralmeas	extremal dependency inference
Stable tail dependence	$\hat{l}(x,y)$	stabtaildepf	extremal dependency inference
L-comoments (samp. distr.)	$--$	lcomCOPpv	experimental bivariate inference

The Table of Probabilities that follows lists important relations between various joint probability concepts, the copula, nonexceedance probabilities $u$ and $v$, and exceedance probabilities $u^{\prime }$ and $v^{\prime }$. A compact summary of these probability relations has obvious usefulness. The notation $[\dots ,\dots ]$ is to read as $[\dots \mathrm{and}\dots ]$, and the $[\dots \mid \dots ]$ is to be read as $[\dots \mathrm{given}\dots ]$.

Probability	and	Symbol Convention
$\mathrm{Pr}[U\le u,V\le v]$	$=$	$\mathbf{C}(u,v)$ --- The copula, COP
$\mathrm{Pr}[U>u,V>v]$	$=$	$\hat{\mathbf{C}}(u^{\prime },v^{\prime })$ --- The survival copula, surCOP
$\mathrm{Pr}[U\le u,V>v]$	$=$	$u-\mathbf{C}(u,v^{\prime })$
$\mathrm{Pr}[U>u,V\le v]$	$=$	$v-\mathbf{C}(u^{\prime },v)$
$\mathrm{Pr}[U\le u\mid V\le v]$	$=$	$\mathbf{C}(u,v)/v$
$\mathrm{Pr}[V\le v\mid U\le u]$	$=$	$\mathbf{C}(u,v)/u$
$\mathrm{Pr}[U\le u\mid V>v]$	$=$	$(u-\mathbf{C}(u,v))/(1-v)$
$\mathrm{Pr}[V\le v\mid U>u]$	$=$	$(v-\mathbf{C}(u,v))/(1-u)$
$\mathrm{Pr}[U>u\mid V>v]$	$=$	$\hat{\mathbf{C}}(u^{\prime },v^{\prime })/u^{\prime }=\bar{\mathbf{C}}(u,v)/(1-u)$
$\mathrm{Pr}[V>v\mid U>u]$	$=$	$\hat{\mathbf{C}}(u^{\prime },v^{\prime })/v^{\prime }=\bar{\mathbf{C}}(u,v)/(1-v)$
$\mathrm{Pr}[V\le v\mid U=u]$	$=$	$\delta \mathbf{C}(u,v)/\delta u$ --- Partial derivative, derCOP
$\mathrm{Pr}[U\le u\mid V=v]$	$=$	$\delta \mathbf{C}(u,v)/\delta v$ --- Partial derivative, derCOP2
$\mathrm{Pr}[U>u\mathrm{or}V>v]$	$=$	${\mathbf{C}}^{\star }(u^{\prime },v^{\prime })=1-\mathbf{C}(u^{\prime },v^{\prime })$ --- The co-copula, coCOP
$\mathrm{Pr}[U\le u\mathrm{or}V\le v]$	$=$	$\tilde{\mathbf{C}}(u,v)=u+v-\mathbf{C}(u,v)$ --- The dual of a copula, duCOP
$E[U\mid V=v]$	$=$	${\int }_0^1(1-\delta \mathbf{C}(u,v)/\delta v)\mathrm{d}u$ --- Expectation of U given V, EuvCOP
$E[V\mid U=u]$	$=$	${\int }_0^1(1-\delta \mathbf{C}(u,v)/\delta u)\mathrm{d}v$ --- Expectation of V given U, EvuCOP

The function jointCOP has considerable demonstration in its Note section of the joint and and joint or relations shown through simulation and counting scenarios. Also there is a demonstration in the Note section of function duCOP on application of the concepts of joint and conditions, joint or conditions, and importantly joint mutually exclusive or conditions.

Copula Construction Methods

Permutation asymmetry can be added to a copula by breveCOP. One, two, or more copulas can be “composited,” “combined,” or “multiplied” in interesting ways to create highly unique bivariate relations and as a result, complex dependence structures can be formed. The package provides three main functions for copula composition: composite1COP composites a single copula with two compositing parameters (Khoudraji device with independence), composite2COP (Khoudraji device) composites two copulas with two compositing parameters, and composite3COP composites two copulas with four compositing parameters. Also two copulas can be combined through a weighted convex combination using convex2COP with a single weighting parameter, and even $N$ number of copulas can be combined by weights using convexCOP. So-called “gluing” two copula by a parameter is provided by glueCOP. Multiplication of two copulas to form a third is supported by prod2COP. All eight functions for compositing, combining, or multipling copulas are compatible with joint probability simulation (simCOP), measures of association (e.g. ${\rho }_{\mathbf{C}}$), and presumably all other copula operations using copBasic features. Finally, ordinal sums of copula are provided by ORDSUMcop and ORDSUWcop as particularly interesting methods of combining copulas.

No. of copulas	Combining Parameters	Function	Concept
1	$\beta$	breveCOP	adding permuation asymmetry
1	$\alpha ,\beta$	composite1COP	copula combination
1	$\alpha ,\beta$	khoudraji1COP	copula combination
1	$\alpha ,\beta$	khoudrajiPCOP	copula combination
2	$\alpha ,\beta$	composite2COP	copula combination
2	$\alpha ,\beta$	khoudraji2COP	copula combination
2	$\alpha ,\beta ,\kappa ,\gamma$	composite3COP	copula combination
2	$\alpha ,(1-\alpha )$	convex2COP	weighted copula combination
$N$	${\omega }_{i\in N}$	convexCOP	weighted copula combination
2	$\gamma$	glueCOP	gluing of coupla
2	$({\mathbf{C}}_1\ast {\mathbf{C}}_2)$	prod2COP	copula multiplication
$N$	${\mathbf{C}}_{\mathcal{J}i}$ for ${\mathcal{J}}_{i\in N}$ partitions	ORDSUMcop	M-ordinal sums of copulas
$N$	${\mathbf{C}}_{\mathcal{J}i}$ for ${\mathcal{J}}_{i\in N}$ partitions	ORDSUWcop	W-ordinal sums of copulas

Useful Copula Relations by Visualization

There are a myriad of relations amongst variables computable through copulas, and these were listed in the Table of Probabilities earlier in this documentation. There is a script located in the inst/doc directory of the copBasic sources titled CopulaRelations_BaseFigure_inR.txt. This script demonstrates, using the PSP copula, relations between the copula (COP), survival copula (surCOP), joint survival function of a copula (surfuncCOP), co-copula (coCOP), and dual of a copula function (duCOP). The script performs simulation and manual counts observations meeting various criteria in order to compute their empirical probabilities. The script produces a base figure, which after extending in editing software, is suitable for educational description and is provided at the end of this documentation.

A Review of “Return Periods” using Copulas

Risk analyses of natural hazards are commonly expressed as annual return periods $T$ in years, which are defined for a nonexceedance probability $q$ as $T=1/(1-q)$. In bivariate analysis, there immediately emerge two types of return periods representing $T_{q;\mathrm{coop}}$ and $T_{q;\mathrm{dual}}$ conditions between nonexceedances of the two hazard sources (random variables) $U$ and $V$. It is usual in many applications for $T$ to be expressed equivalently as a probability $q$ in common for both variables.

Incidentally, the $\mathrm{Pr}[U>u\mid V>v]$ and $\mathrm{Pr}[V>v\mid U>u]$ probabilities also are useful for conditional return period computations following Salvadori et al. (2007, pp. 159--160) but are not further considered here. Also the $F_K(w)$ (Kendall Function or Kendall Measure of a copula) is the core tool for secondary return period computations (see kfuncCOP).

Let the copula $\mathbf{C}(u,v;\mathrm{\Theta })$ for nonexceedances $u$ and $v$ be set for some copula family (formula) by a parameter vector $\mathrm{\Theta }$. The copula family and parameters define the joint coupling (loosely meant the dependency/correlation) between hazards $U$ and $V$. If “failure” occurs if either or both hazards $U$ and $V$ are at probability $q$ threshold ($u=v=1-1/T=q$) for $T$-year return period, then the real return period of failure is defined using either the copula $\mathbf{C}(q,q;\mathrm{\Theta })$ or the co-copula ${\mathbf{C}}^{\star }(q^{\prime },q^{\prime };\mathrm{\Theta })$ for exceedance probability $q^{\prime }=1-q$ is

$$T_{q;\mathrm{coop}}=\frac{1}{1-\mathbf{C}(q,q;\mathrm{\Theta })}=\frac{1}{{\mathbf{C}}^{\star }(1-q,1-q;\mathrm{\Theta })}\text{ and}$$ $$T_{q;\mathrm{coop}}\equiv \frac{1}{\mathrm{cooperative}\mathrm{risk}}\text{.}$$

Or in words, the hazard sources collaborate or cooperate to cause failure. If failure occurs, however, if and only if both hazards $U$ and $V$ occur simultaneously (the hazards must “dually work together” or be “conjunctive”), then the real return period is defined using either the dual of a copula (function) $\tilde{\mathbf{C}}(q,q;\mathrm{\Theta })$, the joint survival function $\bar{\mathbf{C}}(q,q;\mathrm{\Theta })$, or survival copula $\hat{\mathbf{C}}(q^{\prime },q^{\prime };\mathrm{\Theta })$ as

$$T_{q;\mathrm{dual}}=\frac{1}{1-\tilde{\mathbf{C}}(q,q;\mathrm{\Theta })}=\frac{1}{\bar{\mathbf{C}}(q,q;\mathrm{\Theta })}=\frac{1}{\hat{\mathbf{C}}(q^{\prime },q^{\prime };\mathrm{\Theta })}\text{ and}$$ $$T_{q;\mathrm{dual}}\equiv \frac{1}{\mathrm{complement}\mathrm{of}\mathrm{dual}\mathrm{protection}}\text{.}$$

Numerical demonstration is informative. Salvadori et al. (2007, p. 151) show for a Gumbel--Hougaard copula (GHcop) having $\mathrm{\Theta }=$ 3.055 and $T=$ 1,000 years ($q=0.999$) that $T_{q;\mathrm{coop}}=797.1$ years and that $T_{q;\mathrm{dual}}$ = 1,341.4 years, which means that average return periods between “failures” are $$T_{q;\mathrm{coop}}\le T\le T_{q;\mathrm{dual}}\text{ and thus}$$ $$797.1\le T\le 1314.4\text{ years.}$$

With the following code, these bounding return-period values are readily computed and verified using the prob2T() function from the lmomco package along with copBasic functions COP (generic functional interface to a copula) and duCOP (dual of a copula):

  q <- lmomco::T2prob(1000)
  lmomco::prob2T(  COP(q,q, cop=GHcop, para=3.055)) #  797.110
  lmomco::prob2T(duCOP(q,q, cop=GHcop, para=3.055)) # 1341.438

An early source (in 2005) by some of those authors cited on p. 151 of Salvadori et al. (2007; their citation “[67]”) shows $T_{q;\mathrm{dual}}=798$ years---a rounding error seems to have been committed. Finally just for reference, a Gumbel--Hougaard copula having $\mathrm{\Theta }=3.055$ corresponds to an analytical Kendall Tau (see GHcop) of $\tau \approx 0.673$, which can be verified through numerical integration available from tauCOP as:

  tauCOP(cop=GHcop, para=3.055, brute=TRUE) # 0.6726542

Thus, a “better understanding of the statistical characteristics of [multiple hazard sources] requires the study of their joint distribution” (Salvadori et al., 2007, p. 150).

Interaction of copBasic to Copulas in Other Packages

Originally, the copBasic package was not intended to be a port of the numerous bivariate copulas or over re-implementation other bivariate copulas available in R though as the package passed its 10th year in 2018, the original intent changed. It is useful to point out a demonstration showing an implementation of the Gaussian copula from the copula package, which is shown in the Note section of med.regressCOP in a circumstance of ordinary least squares linear regression compared to median regression of a copula as well as prediction limits of both regressions. Another demonstration in context of maximum pseudo-log-likelihood estimation of copula parameters is seen in the Note section mleCOP, and also see “API to the copula package” or “package copula (comparison to)” entries in the Index of this user manual.

Author

William Asquith

References

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

Hernández-Maldonado, V., Díaz-Viera, M., and Erdely, A., 2012, A joint stochastic simulation method using the Bernstein copula as a flexible tool for modeling nonlinear dependence structures between petrophysical properties: Journal of Petroleum Science and Engineering, v. 90--91, pp. 112--123.

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

Hofert, M., Kojadinovic, I., Mächler, M., and Yan, J., 2018, Elements of copula modeling with R: Dordrecht, Netherlands, Springer.

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

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in nature---An approach using copulas: Dordrecht, Netherlands, Springer, Water Science and Technology Library 56, 292 p.