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                NEWS for R Package "copBasic" by W.H. Asquith
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Changes in version 2.2.7 (2025-01-25)
-  Added Frank copula after a few years of trying off and on to determine numerical
     end points determining apparent parameter range suitable for copBasic structure
     and support for Spearman Rho and Kendall Tau. Added example from Joe (2014) using
     the Frank in the context of a C-vine 3-dimensional copula using also the Gumbel. 
     Results show Frank is implemented properly as well as the example of a C-vine.
-  Added additional remarks on the Khoudarji's Devices, which is just the compositing
     of two copula by composite2COP() or with independence in composite1COP().
-  Minor edits to copBasic-package.Rd.
-  Fixed math notation in JOcopB5.Rd.
-  Added keywords "copula (rotation)" and "Theory Examples (reflection/rotation)" to
     applicable *.Rd files.
-  Minor edits in aicCOP.Rd, AMHcop.Rd, asCOP.Rd, copBasic-package.Rd, COP.Rd,
     densityCOP.Rd, densityCOPplot.Rd, jointCOP.Rd, rhoCOP.Rd, tEVcop.Rd, and
     wolfCOP.Rd (confusion fix in one of the Examples).
-  Minor bug fix to kfuncCOP.R where as.sample="Charpentier" was using an undefined
     variable and minor edits to kfuncCOP.Rd.
-  Modified a warning message in each of ORDSUMcop.R and ORDSUWcop.R.
-  Bug fix in Examples for ORDSUWcop.Rd.
-  Grammatical change in PLACKETTsim.Rd.
-  Added switch to M() for large theta in RAYcop().


Changes in version 2.2.6 (2024-10-08)
-  Added Authors@R field to DESCRIPTION as per CRAN requirements.


Changes in version 2.2.5 (2024-10-06)
-  Fixed an error in discussion of the beta range in breveCOP.Rd and minor grammar.
-  Fixed error in math notation for sample Spearman Footrule in footCOP.Rd.
-  Expanded the Note sections in footCOP() and giniCOP() with more sampling theory on
     the two statistics along with adding DOI numbers to two references in each.
-  Minor bug fix in giniCOP(as.sample=TRUE) for division by an integer and not real
     part of the divisor for the sample estimator as per Genest et al. (2010).


Changes in version 2.2.4 (2024-03-06)
-  Added Rayleigh copula as RAYcop().
-  Minor mathematical edits in numerous Rd files; left and right () and [] enlargement
-  Minor grammar edits in numerous Rd files.
-  Error in breveCOP.Rd related to recipes for transforms of breve for optimizations.
-  Added in simCOP.R additional "ditches" to the "dots" to avoid plot() complaining on
     breve argument used by breveCOP().


Changes in version 2.2.3 (2024-01-08)
-  Added additional comments to AMHcop.R, math extensions in AMHcop.Rd, and
     adjustments to Examples therein.
-  Added ordinal sums of W as W_N5p12a() following reproduction of
      Nelsen (2006, exer. 5.12a, p. 172, fig. 5.3a).
-  Added shuffles of M as M_N5p12b() following reproduction of
      Nelsen (2006, exer. 5.12b, p. 173, fig. 5.3b).
-  Added general M-ordinal sums as ORDSUMcop() (Nelsen, 2006, pp. 63-64).
-  Added general W-ordinal sums as ORDSUWcop() (Klement and others, 2017, p. 138).
-  Updated copBasic-package.Rd with ORDSUMcop() entry into a table.
-  Formatting edits in composite1COP.Rd and isCOP.permsym.Rd.
-  Added silent=TRUE default to diagCOPatf(), COPinv(), and COPinv2(). The cascade of
      NAs had become too much of a burden in an otherwise generally working
      algorithm. So, original design ideas from 2008 were changed to silence
      inversion failures.
-  Added the copula of a circular uniform distribution copula as CIRCcop().
-  Added silent=TRUE argument to level.curvesCOP() and level.curvesCOP2() as part of
      silencing try() operations. Also, found nonparallelism between level.curvesCOP()
      and level.curvesCOP2() with the later not having the try() operations. It is
      left in the comments that it is unknown that the try() operation is still
      needed or some deep legacy issue was involved when the EMPIRcop()-operations
      interacted with the level.curvesCOP2().
-  Added bicoploc() as an experimental line-of-organic correlation using inversion
      of a copula. This function is subject to overhaul or even removal but is
      tendered to help a research question by a user.
-  Updates to NAMESPACE to include graphic operations needed within bicoploc()
      diagnostic plots and those functions needed from the lmomco R package.
-  Added a trap in COP() for coupla returned values being > 1 or < 0 to set to the
      closed interval because in unrelated testing a greater than 1 at the
      12th decimal was encountered. Instead of trapping in an application felt best
      to trap in COP(). ** Perhaps other traps might be useful in other functions? **
-  Added argument resamv01 to rCOP() and simCOP() to resample for simulated v == 0 or
      v == 1. The default is false; this new feature maintains the long standing
      behavior of these simulators. An example demonstrating such is added. Added
      argument showresamv01 to display a message of how many of the resampling are
      being done. There is structural protection from infinite looping as well added.
-  Added EvuCOP() and EuvCOP() solving for the expectation of V (U) given U (V),
      respectively with extensive examples and comparison to median regression.
      In general practice, EvuCOP(), which solves E[V | U=u], is expected the more
      common needed.
-  Added EvuCOP() and EuvCOP() to the copBasic-package.Rd.
-  Trapped NaN in a square root of a negative number by truncation errors at the
      end of the PLACKETTcop().
-  Added another simulation example for RFcop.Rd with reflection; revised example
      in W.Rd.
-  Edits to sectionCOP() to prepend the U or V as appropriate to the xlab.
-  Edits in Rds to systematically use latex character "\mid" for "given" made.
-  Edits to convert "Note that" to "Note," for tightness and consistency.
-  Added reporting of attainable Kendall Tau or Spearman Rho values in parameter
      exceedance warnings within AMHcop().
-  Added integrate() control arguments subdivisions, rel.tol, and abs.tol to
      kfuncCOP(), kmeasCOP(), and kfuncCOPinv(). Experiments with secondary return
      period by Kendall Distribution function suggest for some deep-tail estimation
      that additional control of integrate() calls is needed. The defaults are set
      to match those of integrate().
-  Expanded examples and some documentation in kfuncCOPlmoms().


Changes in version 2.2.2 (2023-10-17)
-  Added LzCOPpermsym() as "by far the most interesing" measure of permutation
       asymmetry or exchangability of u,v versus v,u. Features are implemented to
       assist in using LzCOPpermsym() for parameter optimization. This measure is
       named "Permutation-Mu" within the package.
-  Fixed mathematical notation for L_infty^permsym in hoefCOP() as LzCOPpermsym()
       became better understood from one of its references relative to others.
-  Added to aicCOP(), bicCOP, and rmseCOP() ability for first argument to be a matrix
       of the U and V. This is similar to handling in many other copBasic functions.
-  Revised statTn() to have same u,v argument handling as aicCOP(), bicCOP, and
       rmseCOP(). Variable naming fix in one of the Examples in statTn.Rd.
-  Documentation adjustments in asCOP.Rd.
-  Documentation adjustments in breveCOP.Rd.
-  Documentation adjustments in hoefCOP.Rd.
-  Documentation adjustments in isCOP.radsym.
-  Documentation adjustments in copBasic-package.Rd.


Changes in version 2.2.1 (2023-06-21)
-  Extension of breveCOP() to have a breve argument that trumps any inside a para list
       with subordinate extension that a breve inside para trumps beta. This has an
       effect of working more flexibility with composite2COP() and composite3COP().


Changes in version 2.2.0 (2023-06-19)
-  Minor grammar fixes in copBasic-package.Rd.
-  Minor grammar fixes in taildepCOP.Rd and commenting out a print() in taildepCOP.R
-  Cross reference to taildepCOP.Rd made within blomCOPss.Rd.
-  Changes to COP.R with names(para) <- NULL on the returned copula value inheriting
       the names of the parameter and added a reflect <- as.character(reflect) to
       silently permit numerical values (as.integers) for match.arg() and expansion
       of the Examples in COP.Rd. Some other adjustments in the handling of the para
       when it is a list stemming from checks with composite2COP() in the bilmoms()
       and kullCOP() functions. Added example showing the use of the reflection via
       parameter list to the long-standing version of using specific copula
       redefinition options.
-  Changed bilmoms.R and kullCOP.R by adding a scrambling argument for the Sobol
       sequences because randtoolbox::sobol() has disabled the scrambling for which
       copBasic was using scrambling=3 (since version 2.0.2). The default for
       copBasic is scrambling=0, which is the default for randtoolbox::sobol().
       Testing of new operations then required some updates to the Examples in both
       and then some iterative improvement in COP() as it handles para being a
       parameter list.
-  Changed doc/CITATION to use bibentry() as now required by the CRAN.
-  For axes labels involving standard normal scores or variates, a standardization on
       the phrase "STANDARD NORMAL VARIATE OF [U|V]" has been made in
       densityCOPplot.R, EMPIRsim.R, joint.curvesCOP2.Rd, and simCOP.R.
-  For densityCOPplot.R, added snv=TRUE argument with adjustment in how origins=TRUE
       triggers only the origins iff the axes are snv.
-  Tidy up of copBasic/srcall.R for auto directory change using
       rstudioapi::getSourceEditorContext()$path. This function is not part of
       "built" copBasic but only for the developer's use in testing.
-  Extensive grammar editing of all the .Rd files from proofreading the user manual
       and searches and replacings of various terms to build more consistency.
-  Variable renamed from T to TT in diagCOP.R, isCOP.LTD, isCOP.RTI.R, isCOP.permsym.R,
       isCOP.radsym.R, N4212cop.R, PLACKETTcop.R, PLACKETTsim.R, sectionCOP.R,
       so to avoid visual conflict with how syntax markup appears. Such a situation
       does not appear to extend to the letter F.
-  simCOP.R received an as.integer(n) internally to ensure n is an integer to avoid
       warning looking like "user requested n=50.5 simulations" from derCOPinv().
-  Added breveCOP.R/Rd for adding permutation asymmetry to a coupla and this required
       updates to copBasic-package.Rd, minor indexing changes, and cross-linking
       with the other "compositing" coupla (glue, convex, and composite).
-  Added s argument to mleCOP() with the overhaul too of the verbose and messaging
       one can see by simulation the current iteration of the optimizing in U,V
       plotting. This is mostly for teaching purposes but might be useful to many.
-  Added an example in mleCOP.Rd concerning a test using the breveCOP() that has
       been added for this version of the package. Demonstration then made of
       estimation for a simple asymmetric addition to a copula (Plackett choosen).
-  DESCRIPTION has "asymmetry extension" added to it to hint at breveCOP().


Changes in version 2.1.9 (2022-08-30)
-  Incorrect use of \overline in blomCOPss.Rd found during 2.1.8 CRAN submission.


Changes in version 2.1.8 (2022-08-30)
-  Converted the Changelog to this NEWS with reformatting and removed ChangeLog.
-  Added blomatrixCOP() function and made comparisons as part of that development
       to Joe's sample estimator for blomCOP() to the various empirical copula forms
       that have been added (see 2.1.8:3).
-  Added to blomCOP.Rd comparison of Joe's sample estimator to the various
       empirical copula supported by the package. Extended blomCOP() to support the
       empirical copula and did some mathematical edits to the equations shown.
-  Added Blomqvist (Schmid-Schmidt) betas of a copula in blomCOPss().
-  Modified the plotting by taildepCOP() to ensure symmetrical x-axis limits.
-  Extended sample estimation section in Note in taildepCOP.Rd to include use of
       blomCOPss() and show general reliability.
-  From CRAN "check summary", the macros \acute, and \grave, \hat and \overline with
       "new" HTML help building need to be closed with {}. Before, one could write
       \hat\mathbf{C} and get a hat accent not top of a boldface C, but now the
       implicit {} are no longer acceptable to KaTeX instead of LaTeX involved in
       the HTML building. The check error looks like this:
       Unexpected end of input in a macro argument, expected '}' in \hat\mathbf{C}
-   Changes to COP.Rd, copBasic-package.Rd, duCOP.Rd, hoefCOP.Rd, isCOP.radsym.Rd,
       prod2COP.Rd, rhoCOP.Rd, simCOPmicro.Rd, surfuncCOP.Rd, tailordCOP.Rd
       https://blog.r-project.org/2022/04/08/enhancements-to-html-documentation/


Changes in version 2.1.7 (2021-11-14)
-   Minor edits to lcomCOP.Rd and expanded the Examples demonstrating parameter
       estimation for a composite1COP of the Plackett copula.
-   Added cross-reference link to lcomCOP to lcomCOPpv.Rd.
-   Spelling corrections to copBasic-package.Rd and minor augmentations to tables.
-   Minor typesetting edits to jointCOP.Rd.
-   Minor corrections to bilmoms.Rd.
-   Errors in Examples of glueCOP.Rd relative to implementation corrected.
-   Additions/consistency of cross-linking within the composite*COP.Rd, convex*.Rd,
       and glueCOP.Rd files.
-   Corrected minor error on the Raftery copula example in COP.Rd.
-   GHcop.Rd, renamed minus.semicor to botleft.semicor and plus.semicor to
       topright.semicor in preparation for the extension to the other corners in
       the semicorCOP().
-  Added bottom right and top left corners for the semi-correlations in function
       semicorCOP(). This required renaming minus.semicor to botleft.semicor and
       plus.semicor to topright.semicor. The new corners are botright.semicor and
       topleft.semicor. *** There is no backwards compatibility retained to the
       original minus.semicor and plus.semicor names for the returned values. ***
       Overhauled the semicorCOP.Rd and added a four corner example of reflection
       and rotation of the Raftery copula, which ties into the similar example
       of the Raftery in COP.Rd Examples section.
-  Added Zhang and Singh (2019) citation to derCOPinv.Rd as part of replicating
       their example 3.23 for the Clayton copula.
-  Minor edits to prod2COP.Rd, stabtaildepf.Rd, taildepCOP.Rd, and tailordCOP.Rd.
-  Added cross-links to PLcop in PLACKETTpar and PLpar in PLACKETTcop.


Changes in version 2.1.6 (2021-08-30)
-   Minor Rd fixes as R 4.1.+ series emerges and check --as-cran reports new things.
       This mostly added up to the \doi{} call instead of \url{} when a DOI is used.
-   Added a seed argument in simCOP, rCOP, and simCOPmicro to pass to set.seed().
-   Updated the minimum R version through inspection of other dependencies. This
       update is likely not that critical but does bring consistency.


Changes in version 2.1.5 (2020-03-22)
-  Extended RFcop() with fit argument to toggle between Spearman Rho or Kendall Tau.
-  Extended COP.Rd with a example of taking a copula, such as the RFcop, that only
       has positive association and reflecting it three different times to form
       alternative association structures, including negative association, that its
       formula can not natively attain.
-  Audit of AMHcop.Rd found documentation problems in the formula for Spearman's Rho.
       Modified it and AMHcop() to show and respectively use the infinite series of
       Machler (2014). Added the support for rho argument along with a fit argument
       like RFcop() (see 2.1.5:1). Examples in AMHcop.Rd changed.
-  Audit pass of much of the documentation.
-  Changed a few Examples to dontruns because of the maturity of the package. The
       --timings of the R CMD check were consulted. Should be a savings of less than
       one second overall, but trying to help the CRAN.
-  gEVcop.Rd received a minor fix in the Examples.
-  Advancement: EMPIRgrid() was not passing ... to EMPIRcop().
-  Extended EMPIRgridderinv() and EMPIRgridderinv2() to accept a precomputed gridded
       derivative in the dergrid argument.
-  Extended EMPIRgridder() and EMPIRgridder2() to throw a warning about NaN along
       a row or column
         warning("found nonfinite values on row|column=",i," in grid derivative")
       The author has encountered one data set that produces NaN on two rows and
       trying to understand how and why this is happening.
-  Extended the EMPIRcop() with a checker-board empirical copula, which is a type
       of light smooth and adapted from the copula package.
-  Added a trap in EMPIRgridderinv() and EMPIRgridderinv2() if Kumaraswamy is set and
       the Kumaraswamy fit fails, the original approx() to the inversion is
       set for the failure and a warning() triggered as to which index was involved.
-  BUG FIX: An example within EMPIRsim.Rd need fixing and other minor cleanup.
-  Minor spelling checks during Rd auditing for margin protrusion on Rd2pdf.


Changes in version 2.1.4 (2019-07-02)
-  Fixups in FGMcop.Rd (bad URL), HRcop.Rd (no encoding switch in the concept and
       keyword, lcomCOP.Rd (bad URL), rhobevCOP.Rd (no commas in indexing or
       concepts) for release to CRAN.


Changes in version 2.1.3 (2019-07-01)
-  Added documentation in lcomCOP() stating that Chine and Benatia (2017) describe
       trimmed L-comoments as the multivariate extensions of the univariate
       trimmed L-moments. Unknown as to whether an extension jump should be made.
-  Added iterated Farlie-Gumbel-Morgenstern copula as FGMrcop(), which is not any more
       useful than the existing generalization, but Chine and Benatia (2017) concern
       themselves with the iterated FGM.
-  Documentation audit again with tiny changes to many Rds.
-  Added glueCOP() as another means of constructing other copulas.
-  BUG FIX: EMPIRqua.regress() and EMPIRqua.regress2() with lowess=TRUE could return
       probabilities outside of [0,1]. Truncation to zero and unity is now made.
-  Warning Fix: EMPIRgridderinv()/EMPIRgridderinv2() in their calls to approx() could
       trigger warnings about ties and deletion of them. We know therein that the
       x values are ordered, so a switch was made to use ties="ordered" in approx().
       Also, "F" was being used as the nonexceedance probability but R syntax
       highlighting treats as FALSE, so switch made to FF. By association, the
       EMPIRqua.regress()/EMPIRqua.regress2() also need the ties="ordered" in their
       use of the approx() on the inversion of the gridded derivative of the empirical
       copula.
-  Added an as.list() check to the empirical grid argument in both pairs of
       EMPIRgridder()/EMPIRgridder2() and EMPIRgridderinv()/EMPIRgridderinv2().
-  BUG FIX: Through an audit of EMPIRcop.Rd examples, which then found that the
       Bernstein argument to pass through lower-level dispatch to level.curvesCOP()
       did not work. The ability for the para argument in EMPIRcop() now properly
       supports the ability to trigger ctype and the bernstein arguments through
       list embedment.
-  COP.Rd, diagCOPatf.Rd, and med.regressCOP.Rd examples edited to avoid using F with
       change to FF as a variable for nonexceedance probability (see no. 6).


Changes in version 2.1.2 (2019-01-08)
-  Added m=NA argument to aicCOP() and bicCOP() to pass the number of parameters
       of the copula. Trivial to compute the number from standard copulas,
       the problem is how to handle and count on the fly the number of parameters
       for the various the compositing types.
-  Added warning of missing alpha or beta parameters in composite1COP and
       composite2COP. Small speed it, but debugging when these are not given
       (forgotten) becomes complicated. (See #3 and #4)
-  Added warning of missing kappa or gamma parameters in composite3COP.
       (See #2 and #4)
-  Added warning of missing alpha in convex2COP() (See #2 and #3)
-  Enhancement: Added additional indexing for GHcop, GLcop, and HRcop as EV cops.
-  Added tEVcop() with implementation based on the Pickands dependence
       function. Examples are provided. The tEVcop() is deemed useful because
       of interest in semi-exhaustive EV-copula study with the easy addition of
       a 2-parameter copula without triggering convex or compositing. However,
       some literature indicates that the form itself with two parameters is
       not really more expressive than the GHcop, GLcop, and HRcop.
-  Feature fix: COP.R had a weakness when parameters as a list are passed in
       its handling of "cop" versus "cop1", "cop2", etc.
-  BUG FIX: Vectorization in convexCOP.R was not fully working for some
       operations on composited copulas.
-  BUX FIX: isCOP.permsym() and isCOP.radsym() both had dispatching problems
       when composited copulas were involved in "cop" being "matched by
       multiple arguments" error in R. Had this always been a problem or had
       R itself become more restricted? The fix was to insert a COP() call
       as opposed cop() call (the argument named cop).
-  Added the Gaussian extreme-value copula as gEVcop().
-  Added rotation to cophalf (Nelsen's triangular copula) within the examples
       of asCOP.Rd and bilmoms.Rd by including a para argument. This is useful
       for further testing and as related to exploring nuskew() and nustar().
-  Overhauled joeskewCOP() after more extensive testing including nuskew() and
       nustar() for parameter estimation. There are now changes in the numerical
       outputs including the basic scale of each statistic. (See joeskewCOP.Rd)
       The biggest problems seem to have been in the numerical version of the
       sample nustar, which prompted the overhaul. *** No users have reported
       any problems with the older scheme, but it needs to be stated that no
       users are actually known to be using nuskew() and nustar(). ***
-  Added only bilmoms argument to bilmoms() to skip L-comoment computation,
       in order to speed execution.
-  Added lcomCOP() based on integrate(integrate()) nest for computation of the
       bivariate L-moments and then conversion to the L-comoments. This is much
       faster than bilmoms(). It is not remembered by the author why nested
       integration was not originally implemented for bilmoms(). (Sorry.)
       Testing indicates that lcomCOP() works well and is much much faster.
       The function lcomCOPpv() will need revisiting for a future release.
-  Added "1", "2", "3", "4" argument options for reflect in simCOPmicro() and
       simCOPv() to be parallel with COP().
-  Made an editorial sweep throughout the documentation.
-  BUG FIX: semicorCOP() seems to have not had the truncation properly coded,
       which was discovered during inspection of code as the Examples were
       audited. However, the default truncation=0, would not show the problem.
       Signs were not properly being used if truncation > 0 and matching logic
       of Joe (2014, p. 73). Also argument 'samcor' renamed as 'as.sample' to
       be consistent with other functions of the package.
-  Added the Raftery copula as RFcop.
-  GLcop() extended to a two-parameter version as a lower-extreme value
       GL_LEV and related extension to another two-parameter version as the
       "gamma-power mixture" of the Galambos. The two-parameter version does not
       seem materially broader than the one-parameter Galambos (see Note).
       The gamma-power mixture is though a different copula.


Changes in version 2.1.1 (2018-12-01)
-  Enhancement: Changed from sapply to u*v on return. This dates for a long
       time back when the vectorization in R was not idiomatic for the author.
-  Added aicCOP(), bicCOP(), and rmseCOP(): a triad of similar goodness of fits.
-  Minor warning() editing in mleCOP.Rd when alerting user to use interval
       search for one-parameter copulas.
-  Standardized all *.Rd files to utf8 encoding declaration whether needed or no.


Changes in version 2.1.0 (2018-05-07)
-   Minor Rd updates throughout.
-   Revision of warning() in wolfCOP().
-   Removed last of the repeated sequence computation inside nested sapply()s
       with in EMPIRcop(), hoefCOP(), joeskewCOP(), and wolfCOP(). A test wtih
       n=3000 for an as.sample=TRUE computation for wolfCOP() yielded about a
       5-percent decrease in execution time. It is possible that some R idiom
       exists for substantially more speed increase in situations of
       "sapply(..., sapply(...))". However, investigations have not found one and
       use of the "outer()" function does not appear applicable in testing because
       of the lack of symmetry in the matrices involved.
-   Removed repeated sequence computation (though not nested) in kfuncCOP()
       and vuongCOP(). This should be categorized as a "code cleaning" exercise.
-   Fixed error in vuongCOP.Rd about nonrejection of the Plackett with the wording
       should be "rejection." Error was a result of poor cut and paste when first
       written. It was the reverse Gumbel–Hougaard that was not rejected.
-   Fixed error in some accidentally swapped results in Examples of joeskewCOP.Rd.
-   BUG FIX: CLcop() was throwing an error when trying to estimate Kendall Tau and
       the copula parameter from the observed data. The cor(, method="kendall")
       function was being called and the estimate retrieved by $estimate as in
       cor.test()$estimate.
-   Enhancement: Added some parsing of the ... to simCOP on which the delu and
       derdir arguments are identified, used as needed, and then stripped before
       the calls to plot() or points() to avoid "not a graphical parameter"
       warnings. This has been an issue for nearly 10 years but finally WHA
       sat down and figured out how to deal with it.
-   Enhancement: Added a sign test on the objective function in derCOPinv and
       derCOPinv2. This will cut down on repeated warnings during simulations
       when the calls to uniroot breakdown.
-   Added JOcopB5().
-   BUG FIX: rCOP() was not forwarding the sample size on down to simCOPmicro().
-   Added the contour.lwd argument to densityCOPplot().
-   BUG FIX: composite1COP() composite2COP() were not passing ... to COP().
       This has not actually been a problem because most of the practical
       applications of copula herein are not often called with other arguments.
       However, composite3COP() was passing ... down to composite2COP().
-   Added FGMcop() for the Farlie-Gumbel-Morgenstern copula, which is actually
       a recent (2012) generalization of this copula.
-   Added AMHcop() for the Ali-Mikhail-Haq copula.
-   Added alternative reflection arguments "1,2,3,4" to COP().
-   Added alias PAcop() for PARETOcop() to be parallel to the PLACKETTcop().


Changes in version 2.0.7 (2018-03-08)
-   Clarified some documentation in bilmoms.Rd.
-   Expanded description in Examples in kullCOP.Rd.


Changes in version 2.0.6 (2017-11-11)
-   Added convexCOP() as a N-bivariate copula generalization of convex2COP().
-   Documentation adjustment to GHcop.Rd.
-   Added CLcop() for user R. Adat related to his own coding efforts.


Changes in version 2.0.5 (2017-02-24)
-   Typo correction to DESCRIPTION and line ending changes to help hyphenation.
-   NAME CHANGE: StableTailDepFN() --> stabtaildepf()
-   Added importFrom("graphics", "mtext") for stabtaildepf() needs in NAMESPACE.
-   Small adjustments to throughout documentation, but more importantly, tried
       running all examples + "dontrun" declarations + as well as the multitude
       of "preformatted" code snippets. All seem to be working and several
       errors and consistency issues have been resolved---so this exercise was
       reasonably needed.
-   Error in blomCOP.Rd on old argument change: "sambeta" --> "as.sample".
-   NAME ADDITION: PLcop() and PLpar() are now available for Plackett copula.
-   Further auditing of kfuncCOP(), kfuncCOPinv(), and kfuncCOPlmoms(). Expanded
       example of Kendall function L-moments.


Changes in version 2.0.4 (2016-06-03)
-   Microchanges 2.0.3 to pass CRAN importation.


Changes in version 2.0.3 (2016-05-31)
-   Another deep rereading of the documentation and editorial revisions.
-   Extended the Gaussian copula example within med.regressCOP.Rd that deals with the
       interaction with the copula package. (Reminder, WHA does not want to just dump
       other copulas within copBasic. WHA wants just enough directly implemented to
       show and validate core bivariate copula mathematical theory.)
-   Added the.zero and sumlogs to densityCOP() to facilitate maximum likelihood
       estimation (see next item).
-   Added the.zero to vuongCOP() so it can dispatch that to densityCOP().
-   Added mleCOP() as an example of maximum likelihood estimation suitable for
       the copBasic package but Examples extend into how the copula package
       might be integrated to gain access to other copulas but still retain
       a hosting within the copBasic package. Thanks to S. Kloibhofer for ideas.
-   Added rCOP() as a light-weight implementation of simCOP().
-   Extended use of the "tol" argument in taildepCOP() and tailconCOP() to
       control the internally defined "resolution" variable.
-   Added psepolar() for pseudo-polar representation of bivariate data.
-   Added spectralmeas() for the spectral measure distribution of bivariate data
       derived from the pseudo-polar representation.
-   Added StableTailDepFN() for the estimation of the stable tail dependence
       function and construction of level sets associated with it.
-   REMOVED: PlackettPlackettABKGtest.Rd, PlackettPlackettNP.Rd,
             PlackettPlackettABKGtest.RData, PlackettPlackettNP.RData in
       effort to reduce file size foot print of the package.
       See lcomoms2.ABcop2parameterm() and lcomoms2.ABKGcop2parameter() to
       explore how to prepopulate parameter to L-comoment mappings.


Version 2.0.2 (2015-10-31)
-   Added sobol argument to kullCOP() for Sobol sequences in the Monte Carlo
      integration with an example in kullCOP.Rd. The package randtoolbox is
      now imported from to get the sobol() function.
-   Added bilmoms() function to compute bivariate L-moments and remapping them
      to L-comoments of L-correlation, L-coskew, and L-cokurtosis using
      Monte Carlo integration and optional Sobol sequences.
-   Added the two-parameter Gumbel-Hougaard copula that is considered in the
      paper providing the bivariate L-moments.
-   Added level to med.regress() and med.regressCOP2() for prediction interval
      computation and added example to their use involving the Gaussian copula
      from the copula package along with linear regression.
-   Migration of lcomCOPpv() from large sample simulation to Monte Carlo
      integration for population estimate of L-comoments via the new bilmoms().


Changes in version 2.0.1 (2015-08-31)
-  Some major documentation expansion here and there as cascading impacts from
      the other changes listed below. Minor documentation formatting changes in
      a select few Rd files and many proofreading passes through all
      documentation resulted in modest additions and corrections. Overhaul of
      Examples sections to keep --timings down to < 0.01 elapsed seconds
      for all documented functions outside \dontrun{} declarations on test
      platform.
-  Added the "tol" argument to diagCOPatf and specified a tolerance tighter
      than the default of uniroot because of apparent loss of precision
      when experimenting with 1,000,000-year return periods.
-  Added a Note to GHcop.Rd showing analytical derivatives and the inverse.
-  Added jointCOP(), joint.curvesCOP(), joint.curvesCOP2() for an alternative
      framework for computing joint probabilities with a feature of the two
      conditions of AND and OR for the joint probability.
-  Typo correction in copBasic-package.Rd and extensions to that Rd considering
      more about joint AND, joint OR, and mutually exclusive OR probabilities.
-  Added exp(log(...)) logic to PSP() for more accuracy for very small
      probabilities to avoid division by small numbers.
-  Added in PSP.Rd discussion of PSP() returning NaN for zero probabilities.
-  Added (related to no. 5 and 6) trapped against a coupla NaN for zero
      probabilities in root-solving in COPinv() and COPinv2(). The functions
      jointCOP(), joint.curvesCOP(), joint.curvesCOP2() have this trap too.
-  Added Note section in duCOP.Rd about joint AND, OR, and exclusive OR.
-  Added reflection permutations to COP() and simCOPmicro() after Joe (2014) as
      cited in documentation. WHA does not desire to implement the reflection
      feature throughout the package---see the combined documentation of these
      two functions.
-  Changed the named argument "swap" to "wrtV" in qua.regressCOP.draw to
      parallel syntax of similar uses in copBasic.
-  Added inst/doc/CopulaRelationBaseFiginR.txt, which is an R script that
      creates a base graphic suitable for manual vector editing defining
      bivariate relations of probability a copulas.
-  Added Joe's Skewness of a copula by joeskewCOP() after Joe (2014) for
      measurement of permutation asymmetry (skew).
-  Added "RhoE" of a bivariate extreme value copula by rhobevCOP() after
      Joe (2014).
-  Added Kendall Function [kfuncCOP()/kmeasCOP()] along with very considerable
      documentation concerning how they are presented in Nelsen (2006),
      Salvadori et al. (2007), and Joe (2014). The inversion is kfuncCOPinv().
-  Added convexCOP() for a convex combination of two copulas after Joe (2014).
-  Removed the extensive copying of internal results within composite1COP(),
      composite2COP(), and composite3COP()---should be big speed increase for
      giant sample sizes.
-  Added Spearman's Footrule as footCOP() with sample estimation as well.
-  Added sample estimation to Gini Gamma in giniCOP(), Hoeffding Phi in
      hoefCOP(), and Schweizer and Wolff Sigma in wolfCOP(), and for
      parallelism---sample interface in rhoCOP() and tauCOP() to R's cor().
-  Added L-moments of the kfuncCOP() via kfuncCOPlmoms() and kfuncCOPlmom(),
      which closely parallel theoretical L-moment computation using the
      cdf2lmoms() and cdf2lmom() functions of the lmomco package.
-  Added Linear Spearman copula in FRECHETcop().
-  Added Asymmetric Gumbel-Hougaard extension to GHcop() with L-comoment demo.
-  Added prod2COP() for the *-product multiplication of copulas.
-  **DEPRECTATION ALERT** Added Hazen extension to EMPIRcop() and lumped
      computation types into new argument "ctype"" and fully **removed**
      the "weibull"" and "bernstein"" arguments.
-  Added GLcop() [Galambos] and HRcop() [Husler-Reiss] extreme value copulas.
-  **DEPRECTATION ALERT** Overhaul of kullCOP() into Monte Carlo integration
      and deprecated numerical integration to kullCOPint(). Logic is simpler
      at expense of longer computation times but does not suffer from
      hard-to-mitigate divergent integral errors.
-  Added isfuncCOP() as a gridded numerical test on whether a bivariate
      function appears to be a copula. Credit to user S. Kloibhofer for idea
      and initial code---thanks for the support!
-  Added statTn() as a measure of a fitted coupla divergence from the
      empirical copula.
-  Added the tail concentration function in tailconCOP().
-  Added the copBasic.fitpara.beta() in copBasic.fitpara as a potential
      uni-dimensional or multi-dimensional parameter optimization interface.
      This is not a maximum likelihood approach but matching of sample
      statistics to theoretical (e.g. like a method of moments). Discussions
      of maximum likelihood implementation with S. Kloibhofer are ongoing.
-  Overhaul of the output structure of vuongCOP() and added elements on the
      return list to enhance clarity and added t-dist. for confidence interval.


Changes in version 1.7.1 (2015-01-31)
-  Minor changes to CITATION following CRAN email dated 01/09/2015.


Changes in version 1.7.0 (2015-01-03)
-   Further refinement of documentation and continued work on highlighting
      and extending the connections to numerical results of various kinds
      reported in Nelsen (2006) as well as Joe (2014).
-   Extended hoefCOP() into Lp distances from independence and weaved in more
      of Nelsen (2006).
-   Added LpCOP(), which is functionally the same as hoefCOP().
-   Minor documentation tweaks to EMPIRcop.Rd, EMPIRcopdf.Rd, EMPIRgrid.Rd,
      PLACKETTpar.Rd, PlackettPlackettABKGtest.Rd, PlackettPlackettNP.Rd,
      composite1COP.Rd, composite2COP.Rd, composite3COP.Rd,
      isCOP.LTD.Rd, isCOP.RTI.Rd, isCOP.radsym.Rd,
      lcomoms2.ABKGcop2parameter.Rd, and lcomoms2.ABcop2parameter.Rd.
-   Added extended example for isCOP* functions inside isCOP.radsym.Rd.
-   Added the Gumbel-Hougaard copula (GHcop) because WHA is studying the
      method of L-comoments on this copula relative to the Plackett copula.
-   Fixed spelling of Roger Nelsen's last name in all Rd files (sorry Roger).
-   Added another example to copBasic-package.Rd.
-   Added FRECHETcop() and added utf-8 encoding to those Rd files requiring
      the diacritic on Frechet: GHcop.Rd, M.Rd, PlackettPlackettABKGtest.Rd,
      PlackettPlackettNP.Rd, and W.Rd.
-  Enhanced warning message in tauCOP() to point out that some failure of
      numerical integration can be (often is?) attributable to singularity
      for the given copula.
-  Added an example to PLACKETTpar.Rd that shows the development of a
      polynomial for estimation of Kendall's Tau given Spearman's Rho of a
      Plackett copula. Further, PLACKETTpar() now sets labels for the returned
      theta using names().
-  Added additional discussion to isCOP.radsym.Rd to suggest that is seems
      that radially symmetric copulas have zero L-comoments beyond
      L-correlation. The literature seems silent on this question.
-  Improved a conditional on infinity in PLACKETTcop.R.
-  Added 'graphics' argument to simCOP.R to make is easier to turn off
      graphical output from that function and also added the arguments 'snv',
      'infsnv.rm', and 'trapinfsnv' as influence of Joe (2014) is incorporated.
-  Changed to more idiomatic R syntax in derCOPinv.R and derCOPinv2.R so that
      the lines of code were reduced.
-  Added L-comoment example into the independence copula P.Rd.
-  Added diagCOPatf() for solution of the probabilities u and v that provide
      joint probability F.
-  Added densityCOP() and densityCOPplot() following review of Joe (2014).
-  Added sample Blomqvist's Beta computation to blomCOP() after Joe (2014).
-  Added uvlmoms() and uvskewnewss() after Joe (2014).
-  Added lcomCOPpv() for simulating the sample distribution(s) of
      L-correlation, L-coskew, and L-cokurtosis for a copula.
-  Added 'weibull' argument to EMPIRcop() [note the Joe (2014) often uses
      Hazen plotting positions without so naming them as Hazen]
-  Added the Kullback-Leibler divergences, Jeffrey's divergence, and the
      Kullback-Leibler sample size.
-  Added semicorCOP() for semicorrelations of a copula and(or) data.
-  Added isCOP.permsym() for a grid-wise check on permutation symmetry.
-  Added vuongCOP() for Vuong's procedure of copula comparison along with an
      extensive Note section showing the forefront of copBasic extension into
      the inference problem.


Changes in version 1.6.0 (2014-09-26)
-   Added the Bernstein copula extension to EMPIRcop().
-   Changes to derCOP() and derCOP2() of accommodate vectorization yet still
      retain the dynamic suspension of requested derivative direction should
      need to be changed (overrided) [but only when needed]. Thus, tauCOP()
      now does not kick off warnings.
-   Added isCOP.radsym() for a grid-wise check on radial/reflection symmetry.
-   Added asCOP() to help build copulas with operational similarity to the
      expected implementation style for the package.
-   Added isCOP.PQD() for testing positively quadrant dependency of a copula.
-   Added gridCOP() for supporting 3-D or 2-D image renderings of a copula.
-   Enhanced simCOP() for more speed and reporting features and vectorized
      simCOPmicro().
-   Added the surfuncCOP() for the joint survival function of a copula, which
      is subtly different from the survival copula in surCOP.
-   Removed the ratcop argument from sectionCOP(). That feature is too much
      like what isCOP.LTD() or isCOP.RTI() was already doing.
-  Comprehensive Rd auditing including the Examples and example expansion.
-  Removed the capital "F" for quantile regression level and replaced with
      lower case "f" in qua.regress(), qua.regress2(), and updated all
      examples. If I have broken code for a few users, I am sorry but the
      refrain from capitals in function arguments should have been kept
      consistent with version 1.5.0.
-  Added Hoeffding's Phi in hoefCOP().
-  Adjustments to the features of sectionCOP().
-  Enhanced indexing by suggestion of Amaranthe R. Biessen, whom also was
      influential my major index extensions to the lmomco package.
-  Adjusted DESCRIPTION and NAMESPACE to only partially grab those few
      functions needed from the lmomco package.


Changes in version 1.5.4 (2013-05-10)
-  Prof. Ripley remarked that the DESCRIPTION of copBasic was too long.
      This is a reasonably critique. The DESCRIPTION has been radically
      shortened. The original (with new edits and organization) is now located
      in the inst/doc/Extended_DESCRIPTION.txt file. WHA will use this file
      as needed to provide further guidance into the package.
-  Contraction of --timings made to the following: tauCOP.Rd wolfCOP.Rd
      level.curvesCOP.Rd med.regressCOP.Rd qua.regressCOP.Rd qua.regressCOP2.Rd.
      Also some minor tweaks to the median and quantile regression examples were
      made.


Changes in version 1.5.3 (2014-04-20)
-   Added the ReineckeWells.RData and ReineckeWell266.RData.
    ** This version failed --timings checks by CRAN, never released **


Changes in version 1.5.2 (2012-07-30)
-   med.regressCOP() and med.regressCOP() had arguments changed to match
      the qua.regressCOP() and qua.regressCOP2() functions.
-   med.regressCOP.Rd and med.regressCOP2.Rd changed in argument to match code.
-   qua.regressCOP(), qua.regressCOP2(), med.regressCOP(), and med.regressCOP2()
      all now insert an NA in case a root solution could not be found.
-   Spelling corrections in DESCRIPTION: derivative and empirical.
      Thanks CRAN team for the automated spell checkers.


Changes in version 1.5.1 (2011-11-14)
-   Added data/PlackettPlackettABKGtest.RData and
           man/PlackettPlackettABKGtest.Rd.
-   Removed lcomoms2.cop2parameter.R/Rd.
-   Added lcomoms2.ABcop2parameter.R/Rd (replaces lcomoms2.cop2parameter.R/Rd).
-   Added lcomoms2.ABKGcop2parameter.R/Rd.
-   BUG FIX: Labeling inconsistency when sectionCOP() called with dercop=FALSE.
-   Added the EMPIRcop() function for estimation of the empirical copula.
-   Added the EMPIRcopdf() function to compute the entire data.frame() of
      the empirical copula---the copula for each coordinate is computed.
-   Added the EMPIRgrid() function for computation of a regular grid of
      the empirical copula---this grid will form the basis of other copula
      operations on the empirical copula.
-   Added the EMPIRgridder() and EMPIRgridder2() functions for
      computation of the sectional derivatives of the empirical copula from
      EMPIRgrid().
-  Added the EMPIRgridderinv() and EMPIRgridderinv2() functions for
      computation of the sectional derivatives of the empirical copula from
      EMPIRgrid().
-  Added the EMPIRqua.regress() and EMPIRqua.regress2() functions for
      quantile regression from grid emanating from EMPIRgridderinv() and
      EMPIRgridderinv2().
-  Added the EMPIRmed.regress() and EMPIRmed.regress2() functions for
      median regression from grid emanating from EMPIRgridderinv() and
      EMPIRgridderinv2().
-  Added the EMPIRsim() function for simulation of V given U from the
      empirical copula.
-  Added the EMPIRsimv() function for simulation of V given U (by
      argument) from the empirical copula.
-  Documentation fixes in derCOP2.Rd, lcomoms2.ABcop2parameters.Rd,
      lcomoms2.ABKGcop2parameter.Rd, PlackettPlackettNP.Rd, and
      qua.regressCOP.Rd.
-  Enhanced error handling with COPinv(), COPinv2(), derCOPinv(), and
      derCOPinv2().
-  Enhancement to derCOP() and derCOP2() by changing derivative direction
      if near the 0 or 1 boundaries.


Changes in version 1.5.0 (2011-08-31)
-   Enhancement on derCOPinv.R when a solution is unbounded in uniroot()
      to return NA as needed. This was discovered in massive simulation
      from simcompositeCOP()---massive means 500k simulation counts.
-   Added Nelsen's copula in equation 4.2.12 as the N4212cop() function.
-   Removed and restructured the tabulated L-comoment solutions to the
      Plackett-Plackett copula to clean up some loose ends and set up the
      ability to load much much larger solutions sets that are too big
      for CRAN distribution.
-   Major construction of the data subdirectory and changes to documentation.
      Removed comp2lcomsPlackettPlackett.Rd, comp2lcomsPlackettPlackett2.Rd,
      lcom2PLACKETTPLACKETTcop.R/Rd.
-   Added lcomoms2.cop2parameter.R/Rd (replaces lcom2PLACKETTPLACKETTcop.R/Rd).
-   Added data/PlackettPlackettNP.RData and man/PlackettPlackettNP.Rd.
-   Changes to the CITATION file.
-   Removed capitalization of arguments in the empricial quantile regression
      (including f --> f.lowess): EMPIRqua.regress.R, EMPIRqua.regress2.R.
-   Removed capitalization of arguments in the median regression:
      med.regressCOP(), med.regressCOP2(), and
      qua.regressCOP(), qua.regressCOP2(), qua.regressCOP.draw()
-  Added level.setCOP() and level.setCOP2() that are dispatchers to special
      features of level.curvesCOP() and level.curvesCOP2(), respectively. The
      level curves functions required some extensive reorganization and
      further error trapping to support the single level set return.
-  Added surfuncCOP() as the joint survival function. This function is
      closely related to surCOP() but with subtle argument differences. This
      new function included to be ever more consistent with Nelsen's book.
-  Added kmeasCOP() for the Kendall's Measure of a copula, which is critically
      important for "secondary return period" computations.
-  Added alias of simCOPv() for simCOPmicro() to be in parallel with EMPIRsimv.
-  Relations between COP(), surCOP(), surfuncCOP(), duCOP(), and coCOP() are
      illustrated in a script in inst/doc/CopulaRelations_BaseFigure_inR.txt
      as a base figure and vector edited in inst/doc/CopulaRelationsFigure.pdf

Changes in version 1.4 (2008-10-30)
-  Initial release (class project)


--------------------------------------------------------------------------------------

POTENTIAL TODOs:

Continued inspection of the approx() use and whether ties="ordered" can be used
(see 2.1.3#6), it appears in all other cases that yes, we can move
to the ties="ordered" because the inversion of the gridded derivative of the empirical
copula is being used and structurally the "x" part of the approx will be ordered.
R/EMPIRsim.R:      v <- approx(cols, y=v.available, xout=t, rule=2)$y
R/EMPIRsim.R:      v1 <- approx(cols, y=v.available1, xout=t, rule=2)$y
R/EMPIRsim.R:      v2 <- approx(cols, y=v.available2, xout=t, rule=2)$y
R/EMPIRsimv.R:      v <- approx(cols, y=v.available, xout=t, rule=2)$y
R/EMPIRsimv.R:      v1 <- approx(cols, y=v.available1, xout=t, rule=2)$y
R/EMPIRsimv.R:      v2 <- approx(cols, y=v.available2, xout=t, rule=2)$y
R/kfuncCOP.R:         empkc <- approx(c(0,FKin,1), c(0,Zin,1), xout=z)$y


lcomCOPpv() <--> vuongCOP() work as now lcomCOP() is available.
lcomCOPpv() might need some reworking too.

If two-parameter Galambos (Joe, 2014, p. 198) is not materially broader than
the one-parameter, does this hold for a two-parameter Gumbel hinted by
Joe (2014, p. 199)? Further investigation is needed.

Note: Joe's 2-p GHcop does appear different from Brahimi et. al (2015) version of a
2-p GHcop. Should support for both be made? Are both the same? (Quick testing
suggests that perhaps they are not, but mathematical checking has not been made
as of January 2019]).

Add the trimmed L-comoments?

EMPIRcopdf() can be replaced by EMPIRcop() as the later is fully
  vectorized itself and makes presumably a lesser impact on memory (faster)?

stabtaildepf.Rd --- Need to audit stabtaildepf.Rd and complete documentation.
Beirlant, J., Escobar-Bach, M., Goegebeur, Y., Guillou, A.: Bias-corrected
estimation of stable tail dependence function. J. Multivar. Anal. 143, 453–466 (2016)
https://doi.org/10.1016/j.jmva.2015.10.006
Kiriliouk, A., Segers, J. & Tafakori, L. Extremes (2018) 21: 581.
https://doi.org/10.1007/s10687-018-0315-y

Implementation of state-of-the-practice nonparametric estimators of
  lower and upper tail dependence parameters. Neither of the three books
  cited within deal with this. A turn towards recommended literature by
  Joe (2014) is needed.

Goodness-of-fit of the Kendall Function following Genest et. al. (2006,
  Goodness-of-fit procedures for copula models based on the probability
  integral transformation: Scand J. Statist 33.

Convergence evaluators for the Monte Carlo integration in bilmoms() and
  kullCOP() to enhance computation times.

rhobevCOP() study for a copula yielding 0.
