Basic Theoretical Copula, Empirical Copula, and Various Utility
Functions
Description
This package implements extensive, but select, functions for copula computations and is used by several other packages by the author. This particular package provides the lower (W), upper (M), and product (P) copulas as well as the product-summation-product copula (PSP). Wrapper functions for the copula (COP), survival copula (surCOP), dual of a copula (duCOP), and co-copula (coCOP) are provided. The package has (level.curvesCOP) for drawing level curves of a given copula, and this function uses the inverse of a copula (COPinv and a COPinv2 is provided for completeness). The numerical derivative for the derivative of a copula is provided by (derCOP and derCOP2) and the inversion of these functions are by (derCOPinv and derCOPinv2). The diagonal sections of a copula are supported by (diagCOP), and sections and derivatives of sections are supported by (sectionCOP). The inverses of copula derivatives are important for random variate generation. Random variate generation for a copula using the conditional distribution method and the deriviative of a copula is provided by (simCOP and the reduced capability simCOPmicro). For slightly broader application for purposes of education and experimentation with copulas, this package also supports the Plackett copula (PLACKETTcop) because of the general applicability of this copula. The Plackett copula is comprehensive, which means that it can attain complete negative association, independence, and positive association. Plackett parameter estimation is straightforward with (PLACKETTpar). A Plackett-specific, random-variate algorithm is by (PLACKETTsim). Composition of a single copula for two external parameters is by (composite1COP). Composition of two copulas through use of two external parameters is provided by (composite2COP), and the composition of two copulas through the use of four external parameters is provided by (composite3COP). Composite copula random variates are generated by (simcompositeCOP). These compositions generally yield asymmetric copulas. A data set is provided that contains darts thrown at the L-comoment space of a Plackett-Plackett composited compula; these data might be used for experimental copula estimation by the method of L-comoments. The package also provides a full suite of functions to numerically compute measures of association through concordance for a given copula such as Kendall's Tau (tauCOP), Spearman's Rho (rhoCOP), Gini's Gamma (giniCOP), and Blomqvist's Beta (blomCOP). The Schweizer and Wolff's Sigma (wolfCOP) is implemented as a measure of dependency as opposed the the concordance measures just listed. Upper and lower tail dependence is computed by numerical limit convergence by (taildepCOP). A numerical computation of the logical whether a copula is left-tail decreasing or right-tail increasing is provided by (isCOP.LTD and isCOP.RTI). The package supports quantile and median regression through (qua.regressCOP, qua.regress2COP, med.regressCOP, med.regress2COP) for a given copula where "2" represents V with respect to U instead of U with respect to V. The regressions can be plotted by (qua.regressCOP.draw). **Empirical Copulas** Empirical copulas are supported by (EMPIRcop) and the computation of a data frame of the copula for each sample value is provided by (EMPIRcopdf). The empirical copula functions are heavily dependent on a simple grid or matrix structure, which is created by (EMPIRgrid). The derivatives of the grid, which are the conditional cumulative distribution functions of the copula sections, are computed by (EMPIRgridder and EMPIRgridder2). The inverses of the derivatives, which are the conditional quantile functions of the copula sections, are computed by (EMPIRgridderinv and EMPIRgridderinv2). Support for median and quantile regression of the empricial copula are provided by (EMPIRmed.regress, EMPIRmed.regress2, EMPIRqua.regress, EMPIRqua.regress2), which use the grids emanating from (EMPIRgridderinv and EMPIRgridderinv2). Support for simulation of V using U from an empirical copula is provided by (EMPIRsim or by EMPIRsimv).