| Type: | Package |
| Title: | Estimation and Goodness-of-Fit of Copula-Based Models with Arbitrary Distributions |
| Version: | 0.5.0 |
| Description: | Estimation and goodness-of-fit functions for copula-based models of bivariate data with arbitrary distributions (discrete, continuous, mixture of both types). The copula families considered here are the Gaussian, Student, Clayton, Frank, Gumbel, Joe, Plackett, BB1, BB6, BB7,BB8, together with the following non-central squared copula families in Nasri (2020) <doi:10.1016/j.spl.2020.108704>: ncs-gaussian, ncs-clayton, ncs-gumbel, ncs-frank, ncs-joe, and ncs-plackett. For theoretical details, see, e.g., Nasri and Remillard (2023) <doi:10.48550/arXiv.2301.13408>. |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| LazyData: | true |
| Depends: | R (≥ 3.5.0), doParallel, parallel, foreach, stats, rvinecopulib, Matrix |
| RoxygenNote: | 7.2.3 |
| NeedsCompilation: | yes |
| Packaged: | 2023-04-20 15:41:28 UTC; Utilisateur |
| Author: | Bouchra R. Nasri [aut, cre, cph], Bruno N Remillard [aut] |
| Maintainer: | Bouchra R. Nasri <bouchra.nasri@umontreal.ca> |
| Repository: | CRAN |
| Date/Publication: | 2023-04-21 07:32:37 UTC |
Auxiliary functions
Description
This function computes the empirical margins, their left-limits, Kendall's tau and Spearman's rho for arbitrary data. Slower than AuxFunC based on C.
Usage
AuxFun(data)
Arguments
data |
Matrix (x,y) of size n x 2 |
Value
tau |
Kendall's tau |
rho |
Spearman's rho |
Fx |
Empirical cdf of x |
Fxm |
Left-limit of the empiricial cdf of x |
Fy |
Empirical cdf of y |
Fym |
Left-limit of the empiricial cdf of y |
References
Nasri (2022). Test of serial dependence for arbitrary distributions. JMVA
Nasri & Remillard (2023). Tests of independence and randomness for arbitrary data using copula-based covariances, arXiv 2301.07267.
Examples
data(simgumbel)
out=AuxFun(simgumbel)
Auxiliary functions using C
Description
This function computes the empirical margins, their left-limits, Kendall's tau and Spearman's rho for arbitrary data
Usage
AuxFunC(data)
Arguments
data |
Matrix (x,y) of size n x 2 |
Value
tau |
Kendall's tau |
rho |
Spearman's rho |
Fx |
Empirical cdf of x |
Fxm |
Left-limit of the empirical cdf of x |
Fy |
Empirical cdf of y |
Fym |
Left-limit of the empirical cdf of y |
References
Nasri (2022). Test of serial dependence for arbitrary distributions. JMVA
Nasri & Remillard (2023). Tests of independence and randomness for arbitrary data using copula-based covariances, arXiv 2301.07267.
Examples
data(simgumbel)
out=AuxFunC(simgumbel)
Empirical bivariate cdf
Description
This function computes the empirical joint cdf evaluated at all points (y1,y2)
Usage
BiEmpCdf(data, y1, y2)
Arguments
data |
Matrix (x1,x2) of size n x 2 |
y1 |
Vector of size n1 |
y2 |
Vector of size n2 |
Value
cdf |
Empirical cdf |
Examples
data(simgumbel)
out=BiEmpCdf(simgumbel,c(0,1),c(-1,0,1))
Quantile function
Description
This function computes the inverse of the cdf of a finite distribution for a vector of probabilities.
Usage
CdfInv(u, y, Fn)
Arguments
u |
Vector of probabilities |
y |
Ordered values |
Fn |
Cdf |
Value
x |
Vector of quantiles |
Examples
y=c(0,1,2)
Fn = c(0.5,0.85,1)
out=CdfInv(c(1:9)/10,y,Fn)
Empirical univariate cdf
Description
This function computes the empirical cdf evaluated at all sample points
Usage
EmpCdf(x)
Arguments
x |
Observations |
Value
Fx |
Empirical cdf |
Fxm |
Left limit of the empirical cdf |
Ix |
Indicator of atoms |
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Examples
data(simgumbel)
out=EmpCdf(simgumbel[,1])
Parameter estimation for bivariate copula-based models with arbitrary distributions
Description
Computes the estimation of the parameters of a copula-based model with arbitrary distributions, i.e, possibly mixtures of discrete and continuous distributions. Parametric margins are allowed. The estimation is based on a pseudo-likelihood adapted to ties.
Usage
EstBiCop(
data = NULL,
family,
rotation = 0,
Fx = NULL,
Fxm = NULL,
Fy = NULL,
Fym = NULL
)
Arguments
data |
Matrix or data frame with 2 columns (X,Y). Can be pseudo-observations. If NULL, Fx and Fy must be provided. |
family |
Copula family: "gaussian", "t", "clayton", "frank", "gumbel", "joe", "plackett”, "bb1", "bb6", "bb7","bb8","ncs-gaussian", "ncs-clayton", "ncs-gumbel", "ncs-frank", "ncs-joe","ncs-plackett". |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
Fx |
Marginal cdf function applied to X (default is NULL). |
Fxm |
Left-limit of marginal cdf function applied to X default is NULL). |
Fy |
Marginal cdf function applied to Y (default is NULL). |
Fym |
Left-limit of marginal cdf function applied to Y (default is NULL). |
Value
par |
Copula parameters |
family |
Copula family |
rotation |
Rotation value |
tauth |
Kendall's tau corresponding to the estimated parameter |
tauemp |
Empirical Kendall's tau (from the multilinear empirical copula) |
rhoSth |
Spearman's rho corresponding to the estimated parameter |
rhoSemp |
Empirical Spearman's tau (from the multilinear empirical copula) |
loglik |
Log-likelihood |
aic |
Aic value |
bic |
Bic value |
data |
Matrix of values (could be (Fx,Fy)) |
F1 |
Cdf of X (Fx if provided, empirical otherwise) |
F1m |
Left-limit of F1 (Fxm if provided, empirical otherwise) |
F2 |
Cdf of Y (Fy if provided, empirical otherwise) |
F2m |
Left-limit of F2 (Fym if provided, empirical otherwise) |
ccdfx |
Conditional cdf of X given Y and it left limit |
ccdfxm |
Left-limit of ccdfx |
ccdfy |
Conditional cdf of Y given X and it left limit |
ccdfym |
Left-limit of ccdfy |
References
Nasri & Remillard (2023). Identifiability and inference for copula-based semiparametric models for random vectors with arbitrary marginal distributions. arXiv 2301.13408.
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
set.seed(2)
data = matrix(rpois(20,1),ncol=2)
out0=EstBiCop(data,"gumbel")
Kendall's tau and Spearman's rho
Description
This function computes Kendall's tau and Spearman's rho for arbitrary data. These are invariant by increasing mappings.
Usage
EstDep(data)
Arguments
data |
Matrix or data frame with 2 columns (X,Y). Can be pseudo-observations. |
Value
tau |
Kendall's tau |
rho |
Spearman's rho |
References
Nasri (2022). Test of serial dependence for arbitrary distributions. JMVA
Nasri & Remillard (2023). Tests of independence and randomness for arbitrary data using copula-based covariances, arXiv 2301.07267.
Examples
data(simgumbel)
out=EstDep(simgumbel)
Quantile function of margins
Description
This function computes the quantile of seven cdf used in Nasri (2022).
Usage
Finv(u, k)
Arguments
u |
Vector of probabilities |
k |
Marginal distribution: [1] Bernoulli(0.8), [2] Poisson(6), [3] Negative binomial with r = 1.5, p = 0.2, [4] Zero-inflated Poisson (10) with w = 0.1 and P(6.67) otherwise, [5] Zero-inflated Gaussian, [6] Discretized Gaussian, [7] Discrete Pareto(1) |
Value
x |
Vector of quantiles |
Author(s)
Bouchra R. Nasri January 2021
References
B.R Nasri (2022). Tests of serial dependence for arbitrary distributions
Examples
x = Finv(runif(40),2)
Goodness-of-fit for bivariate copula-based models with arbitrary distributions
Description
Goodness-of-fit tests for copula-based models for data with arbitrary distributions. The tests statistics are the Cramer-von Mises statistic (Sn), the difference between the empirical Kendall's tau and the theoretical one, and the difference between the empirical Spearman's rho and the theoretical one.
Usage
GofBiCop(
data = NULL,
family,
rotation = 0,
Fx = NULL,
Fxm = NULL,
Fy = NULL,
Fym = NULL,
B = 100,
n_cores = 1
)
Arguments
data |
Matrix or data frame with 2 columns (X,Y). Can be pseudo-observations. If NULL, Fx and Fy must be provided. |
family |
Copula family: "gaussian", "t", "clayton", "frank", "gumbel", "joe", "plackett”, "bb1", "bb6", "bb7","bb8","ncs-gaussian", "ncs-clayton", "ncs-gumbel", "ncs-frank", "ncs-joe","ncs-plackett". |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
Fx |
marginal cdf function applied to X (default is NULL). |
Fxm |
left limit of marginal cdf function applied to X default is NULL). |
Fy |
marginal cdf function applied to Y (default is NULL). |
Fym |
left limit of marginal cdf function applied to Y (default is NULL). |
B |
Number of bootstrap samples (default 100) |
n_cores |
Number of cores to be used for parallel computing (default is 1). |
Value
pvalueSn |
Pvalue of Sn in percent |
pvalueTn |
Pvalue of Tn in percent |
pvalueRn |
Pvalue of Rn in percent |
Sn |
Value of Cramer-von Mises statistic Sn |
Tn |
Value of Kendall's statistic Tn |
Rn |
Value of Spearman's statistic Rn |
cpar |
Copula parameters |
family |
Copula family |
rotation |
Rotation value |
tauth |
Kendall's tau (from the multilinear theoretical copula) |
tauemp |
Empirical Kendall's tau (from the multilinear empirical copula) |
rhoth |
Spearman's rho (from the multilinear theoretical copula) |
rhoemp |
Empirical Spearman's rho (from the multilinear empirical copula) |
parB |
Bootstrapped parameters |
loglik |
Log-likelihood |
aic |
AIC value |
bic |
BIC value |
References
Nasri & Remillard (2023). Identifiability and inference for copula-based semiparametric models for random vectors with arbitrary marginal distributions. arXiv 2301.13408.
Nasri & Remillard (2023). Goodness-of-fit and bootstrapping for copula-based random vectors with arbitrary marginal distributions.
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
data = rvinecopulib::rbicop(10,"gumbel",rotation=0,2)
out=GofBiCop(data,family="gumbel",B=10)
Function to perform parametric bootstrap for goodness-of-fit tests
Description
This function simulates Cramer-von Mises statistic and Kendall's statistic using parametric bootstrap for arbitrary data.
Usage
bootstrapfun(object)
Arguments
object |
object of class 'statcvm'. |
Value
Sn |
Simulated value of the Cramer-von Mises statistic |
Tn |
simulated value of the Kendall's statistic |
Rn |
simulated value of the Spearman's statistic |
parB |
Estimated parameter |
References
Nasri & Remillard (2023). Identifiability and inference for copula-based semiparametric models for random vectors with arbitrary marginal distributions. arXiv 2301.13408.
Nasri & Remillard (2023). Goodness-of-fit and bootstrapping for copula-based random vectors with arbitrary marginal distributions.
Density of non-central squared copula
Description
This function computes the density of the non-central squared copula (ncs) associated with a one-parameter copula with parameter cpar, and parameters a1, a2 >0 .
Usage
dncs(data, family, rotation = 0, par)
Arguments
data |
Matrix (x,y) of size n x 2 |
family |
Copula family: "ncs-gaussian", "ncs-clayton", "ncs-frank", "ncs-gumbel", "ncs-joe", "ncs-plackett”. |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
vector of copula parameter and non-centrality parameter a1,a2 >0 |
Value
pdf |
Density |
References
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
dncs(c(0.5,0.8),"ncs-clayton",par=c(2,1,2))
Density of Plackett copula
Description
This function computes the density of the Plackett copula with parameter par>0.
Usage
dplac(data, rotation = 0, par)
Arguments
data |
Matrix (x,y) of size n x 2 |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
Copula parameter >0 |
Value
pdf |
Density |
Examples
dplac(c(0.5,0.8),par=3,rotation=270)
Options for the estimation of the parameters of bivariate copula-based models
Description
Sets starting values, upper and lower bounds for the parameters. The bounds are based on those in the rvinecopulib package.
Usage
est_options(family, tau = 0.5)
Arguments
family |
Copula family: "gaussian", "t", "clayton", "frank", "gumbel", "joe", "plackett”, "bb1", "bb6", "bb7","bb8","ncs-gaussian", "ncs-clayton", "ncs-gumbel", "ncs-frank", "ncs-joe","ncs-plackett". |
tau |
Estimated Kendall's tau to compute a starting point (default is 0.5) |
Value
LB |
Lower bound for the parameters |
UB |
Upper bound for the parameters |
start |
Starting point for the estimation |
References
Nagler & Vatter (2002). rvinecopulib: High Performance Algorithms for Vine Copula Modeling. Version 0.6.2.1.3
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Nasri (2022). Test of serial dependence for arbitrary distributions. JMVA.
Nasri & Remillard (2023). Copula-based dependence measures for arbitrary data, arXiv 2301.07267.
Examples
out = est_options("bb8")
Family number corresponding to VineCopula package
Description
Computes the number associated with a copula family (without rotation)
Usage
fnumber(family)
Arguments
family |
Copula family: "gaussian", "t", "clayton", "frank", "gumbel", "joe", "plackett”, "bb1", "bb6", "bb7","bb8". |
Value
fnumber |
Number |
References
Nagler et al. (2023). VineCopula: Statistical Inference of Vine Copulas, version 2.4.5.
Examples
fnumber("bb1")
Conditional distribution of non-central squared copula
Description
This function computes the conditional distribution of the non-central squared copula (ncs) associated with a one-parameter copula with parameter cpar, and parameters a1, a2 >0 .
Usage
hncs(data, cond_var, family, rotation = 0, par)
Arguments
data |
Matrix (x,y) of size n x 2 |
cond_var |
Conditioning variable (1 or 2) |
family |
Copula family: "ncs-gaussian", "ncs-clayton", "ncs-frank", "ncs-gumbel", "ncs-joe", "ncs-plackett”. |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
vector of copula parameter and non-centrality parameter a1,a2 >0 |
Value
h |
Conditional cdf |
References
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
hncs(c(0.5,0.8),1,"ncs-clayton",270,c(2,1,2))
Conditional distribution of Plackett copula
Description
This function computes the conditional distribution of the Plackett copula with parameter par>0.
Usage
hplac(data, cond_var, rotation = 0, par)
Arguments
data |
Matrix (x,y) of size n x 2 |
cond_var |
Conditioning variable (1 or 2) |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
Copula parameter >0 |
Value
h |
Conditional cdf |
Examples
hplac(c(0.5,0.8),1,270,3)
Identifiability of two-parameter copula families
Description
Determines if a copula family is identifiable with respect to the empirical margins. One-parameter copula families ("gaussian","gumbel","clayton","frank","plackett","joe") are identifiable whatever the margins. The rank of the gradient of the copula on the range of the margins is evaluated at 10000 parameter points within the lower and upper bounds of the copula family.
Usage
identifiability(data = NULL, family, rotation = 0, Fx = NULL, Fy = NULL)
Arguments
data |
Matrix or data frame with 2 columns (X,Y). Can be pseudo-observations. If NULL, Fx and Fy must be provided. |
family |
Copula family: "gaussian", "t", "clayton", "frank", "gumbel", "joe", "plackett”, "bb1", "bb6", "bb7","bb8","ncs-gaussian", "ncs-clayton", "ncs-gumbel", "ncs-frank", "ncs-joe","ncs-plackett". |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
Fx |
Marginal cdf function applied to X (default is NULL). |
Fy |
Marginal cdf function applied to Y (default is NULL). |
Value
out |
True or False |
References
Nasri & Remillard (2023). Identifiability and inference for copula-based semiparametric models for random vectors with arbitrary marginal distributions. arXiv 2301.13408.
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
set.seed(1)
data = matrix(rpois(20,1),ncol=2)
out = identifiability(data,"gumbel")
Cdf for non-central squared copula
Description
This function computes the distribution function of the non-central squared copula (ncs) associated a with one-parameter copula with parameter cpar, and parameters a1, a2 >0 .
Usage
pncs(data, family, rotation = 0, par)
Arguments
data |
Matrix (x,y) of size n x 2 |
family |
Copula family: "ncs-gaussian", "ncs-clayton", "ncs-frank", "ncs-gumbel", "ncs-joe", "ncs-plackett”. |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
vector of copula parameter and non-centrality parameter a1,a2 >0 |
Value
cdf |
Value of cdf |
References
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
pncs(c(0.5,0.8),"ncs-clayton", par=c(2,1,2),rotation=270)
Cdf for Plackett copula
Description
This function computes the distribution function of the Plackett copula with parameter par>0.
Usage
pplac(data, rotation = 0, par)
Arguments
data |
Matrix (x,y) of size n x 2 |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
Copula parameter >0 |
Value
cdf |
Value of cdf |
Examples
pplac(c(0.5,0.8),270,3)
Computes unique values, cdf and pdf
Description
This function computes the unique values, cdf and pdf for a series of data.
Usage
preparedata(x)
Arguments
x |
Vector |
Value
values |
Unique (sorted) values |
m |
Number of unique values |
Fn |
Empirical cdf of the unique values |
fn |
Empirical pdf of the unique values |
References
B.R. Nasri (2022). Tests of serial dependence for arbitrary distributions
C. Genest, J.G. Neslehova, B.N. Remillard and O. Murphy (2019). Testing for independence in arbitrary distributions.
#'@examples x = c(0,0,0,2,3,1,3,1,2,0) out = prepare_data(x)
Spearman's rho for Plackett copula
Description
Computes the theoretical Spearman's rho for Plackett copula
Usage
rhoplackett(cpar, rotation = 0)
Arguments
cpar |
Copula parameter; can be a vector. |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
Value
rho |
Spearman's rho |
References
Remillard (2013). Statistical Methods for Financial Engineering. CRC Press
Examples
rhoplackett(3,rotation=90)
Simulation of non-central squared copula
Description
This function computes generates a bivariate sample from a non-central squared copula (ncs) associated with a one-parameter copula with parameter cpar, and parameters a1, a2 >0 .
Usage
rncs(n, family, rotation = 0, par)
Arguments
n |
Number of observations |
family |
Copula family: "ncs-gaussian", "ncs-clayton", "ncs-frank", "ncs-gumbel", "ncs-joe", "ncs-plackett”. |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
vector of copula parameter and non-centrality parameter a1,a2 >0 |
Value
U |
Observations |
References
Nasri (2020). On non-central squared copulas. Statistics and Probability Letters.
Examples
rncs(100,"ncs-clayton",par=c(2,1,2))
Generates observations from the Plackett copula
Description
This function generates observations from a Plackett copula with parameter par>0.
Usage
rplac(n, rotation = 0, par)
Arguments
n |
Number of pairs to be generated |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
par |
Copula parameter >0 |
Value
U |
Matrix of observations |
Examples
rplac(10,rotation=90,par=2)
Simulated data
Description
Simulated data from a Gumbel copula with parameter 2, Bernoulli margin for X1 and zero-inflated Gaussian margin for X2.
Usage
data(simgumbel)Format
Data frame of numerical values
Examples
data(simgumbel)
plot(simgumbel,xlab="X1", ylab="X2")
Goodness-of-fit statistics
Description
Computation of goodness-of-fit statistics (Cramer-von Mises and the Kendall's tau)
Usage
statcvm(object)
Arguments
object |
Object of class 'EstBiCop'. |
Value
Sn |
Cramer-von Mises statistic |
Tn |
Kendall's statistic |
Rn |
Spearman's statistic |
tauemp |
Empirical Kendall's tau |
tauth |
Kendall's tau of the multilineat theoretical copula |
rhoemp |
Empirical Spearman's rho |
rhoth |
Spearman's rho of the multilineat theoretical copula |
Y1 |
Ordered observed values of X1 |
F1 |
Empirical cdf of Y1 |
Y2 |
Ordered observed values of X2 |
F2 |
Empirical cdf of Y2 |
cpar |
Copula parameters |
family |
Copula family |
rotation |
Rotation value |
n |
Sample size |
References
Nasri & Remillard (2023). Identifiability and inference for copula-based semiparametric models for random vectors with arbitrary marginal distributions. arXiv 2301.13408.
Examples
set.seed(2)
data = matrix(rpois(20,1),ncol=2)
out0 = EstBiCop(data,"gumbel")
out = statcvm(out0)
Kendall's tau for a copula family
Description
This function computes Kendall's tau for a copula family
Usage
taucop(family_number, cpar, rotation = 0)
Arguments
family_number |
Integer from 1 to 10 |
cpar |
Copula parameters |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
Value
tau |
Kendall's tau |
Examples
taucop(4,2,270) # Gumbel copula
Kendall's tau for Plackettfamily
Description
This function computes Kendall's tau for Plackett family using numerical integration
Usage
tauplackett(cpar, rotation = 0)
Arguments
cpar |
Copula parameter >0 |
rotation |
Rotation: 0 (default value), 90, 180, or 270. |
Value
tau |
Kendall's tau |
Examples
tauplackett(2,270)