Package 'CopulaInference'

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)

---

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)

---

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])

---

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")

---

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)

---

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")

---

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)

---

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)

---