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
data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAASCAYAAABWzo5XAAAAWElEQVR42mNgGPTAxsZmJsVqQApgmGw1yApwKcQiT7phRBuCzzCSDSHGMKINIeDNmWQlA2IigKJwIssQkHdINgxfmBBtGDEBS3KCxBc7pMQgMYE5c/AXPwAwSX4lV3pTWwAAAABJRU5ErkJggg==
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)