Package 'GenOrd' reference manual

Title:	Simulation of Discrete Random Variables with Marginal Distributions and Correlation Matrix and via a Gaussian or Student's t Copula
Description:	A Gaussian or Student's t copula-based procedure for generating samples from discrete random variables with prescribed correlation matrix and marginal distributions.
Authors:	Alessandro Barbiero [aut, cre], Pier Alda Ferrari [aut]
Maintainer:	Alessandro Barbiero <[email protected]>
License:	GPL-3
Version:	2.1.0
Built:	2026-07-03 10:00:38 UTC
Source:	https://github.com/cran/GenOrd

Simulation of Discrete Random Variables with a Given Marginal Distributions and Correlation Matrix

Description

The package implements a procedure for generating samples from a multivariate discrete random variable with pre-specified correlation matrix and marginal distributions. The marginal distributions are linked together through either a Gaussian or a Student's t copula. The procedure is developed in two steps: the first step (function ordcont) sets up the Gaussian/t copula in order to achieve the desired correlation matrix on the target random discrete components; the second step (ordsample) generates samples from the target variables. The procedure can handle both Pearson and Spearman correlations, and any finite support for the discrete variables. The intermediate function contord computes the correlations of the multivariate discrete variable derived from discretization of a multivariate normal or Student's t variable. Function corrcheck returns the lower and upper bounds of the correlation coefficient of each pair of discrete variables given their marginal distributions, i.e., returns the range of feasible bivariate correlations. Function estcontord, to be used with bivariate samples only, estimates the parameters of the Gaussian/t copula and possibly of the margins.

Compared to version 2.0.0, this version has refined the optimization routine and t probability calculation involved in the estimation process.

Details

Package:	GenOrd
Type:	Package
Version:	2.1.0
Date:	2026-06-30
License:	GPL
LazyLoad:	yes

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Maintainer: Alessandro Barbiero <[email protected]>

References

P.A. Ferrari, A. Barbiero (2012) Simulating ordinal data, Multivariate Behavioral Research, 47(4), 566-589

A. Barbiero, P.A. Ferrari (2015) Simulation of correlated Poisson variables. Applied Stochastic Models in Business and Industry, 31(5), 669-680.

A. Barbiero, P.A. Ferrari (2017) An R package for the simulation of correlated discrete variables. Communications in Statistics-Simulation and Computation, 46(7), 5123-5140.

A. Barbiero, A. Hitaj (2025) Simulating correlated ordinal data via the multivariate Student's t distribution, 2025 6th International Conference on Data Analytics for Business and Industry (ICDABI)

Correlations of discretized variables

Description

The function computes the correlation matrix of the $k$ variables, with given marginal distributions, derived by discretizing a $k$ -variate standard normal or Student's $t$ variable with a given correlation matrix

Usage

contord(marginal, Sigma,
support = list(),
Spearman = FALSE,
df=Inf,
integerdf=TRUE,
prob=FALSE)
contord(marginal, Sigma,
support = list(),
Spearman = FALSE,
df=Inf,
integerdf=TRUE,
prob=FALSE)

Arguments

marginal

a list of $k$ elements, where $k$ is the number of variables. The $i$ -th element of marginal is the vector of the cumulative probabilities defining the marginal distribution of the $i$ -th component of the multivariate variable. If the $i$ -th component can take $k_i$ values, the $i$ -th element of marginal will contain $k_i-1$ probabilities (the $k_i$ -th is obviously 1 and shall not be included).

Sigma

the correlation matrix of the standard multivariate normal variable

support

a list of $k$ elements, where $k$ is the number of variables. The $i$ -th element of support is the vector containing the ordered values of the support of the $i$ -th variable. By default, the support of the $i$ -th variable is $1,2,...,k_i$

Spearman

if TRUE, the function finds Spearman's correlations (and it is not necessary to provide support), if FALSE (default) Pearson's correlations

df

the degrees of freedom of the multivariate Student's $t$

integerdf

if TRUE uses pmvt, which requires an integer value for df; otherwise, uses pmvt.alt, which integrates dmvt with possibly non-integer df

prob

if TRUE, it also returns the probability table in case of a bivariate discrete random variable

Value

The correlation matrix of the discretized variables; if prob=TRUE, it returns a list containing the correlation along with the $k$ -variate probability table

Note

The degrees-of-freedom parameter df can be set to a non-integer value; in this case, the function pmvt.alt is used

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

# Example 1
# consider a bivariate discrete random vector
k <- 2
# with these cumulative margins
marginal <- list(c(1/3, 2/3), c(0.1, 0.3, 0.6))
# generated by discretizing a multivariate standard normal variable
# with correlation matrix
Sigma <- matrix(c(1, .75, .75, 1), 2, 2)
# the resulting joint distribution and correlation matrix
# for the bivariate discrete random vector are
res <- contord(marginal, Sigma, prob=TRUE)
res$pij
res$SigmaO
# let's check if the margins are those assigned
cumsum(margin.table(res$pij,1))
cumsum(margin.table(res$pij,2))
# -> OK
# Example 2
# consider 4 discrete variables
k <- 4
# with these marginal distributions
marginal <- list(0.4,c(0.3,0.6), c(0.25,0.5,0.75), c(0.1,0.2,0.8,0.9))
# generated by discretizing a multivariate standard normal variable
# with correlation matrix
Sigma <- matrix(0.5,4,4)
diag(Sigma) <- 1
# the resulting correlation matrix for the discrete variables is
contord(marginal, Sigma)
# note that all the correlations are smaller than the original 0.5
# change Sigma, adding a negative correlation
Sigma[1,2] <- -0.15
Sigma[2,1] <- Sigma[1,2]
Sigma
# checking whether Sigma is still positive definite
eigen(Sigma)$values # all >0, OK
contord(marginal, Sigma)
# Example 2-bis
# the same margins and the same correlation matrix
# but now we consider a 4-variate Students's t with df=3
contord(marginal, Sigma, df=3)
# -> a slight reduction in magnitude for all the correlations
# Example 3
margin1 <- c(0.2,0.4,0.6,0.8)
margin2 <- margin1
marginal <- list(margin1, margin2)
sigma <- matrix(c(1,0.3,0.3,1),2,2)
df <- 3.5
contord(marginal=marginal, Sigma=sigma, df=df, integerdf=FALSE, prob=TRUE)
# compare with
contord(marginal=marginal, Sigma=sigma, df=round(df), prob=TRUE)
# Example 3
# multivariate probability tables
# Consider a t-copula-based model with these parameters:
margins <- list(1/2, c(1/3,2/3), c(1/6,1/2,5/6))
Sigma <- matrix(0.3, 3 ,3)
diag(Sigma) <- 1
df <- 3.5
# The 3-variate probability table of the discretized distribution is obtained by calling contord
# by setting integerdf=FALSE and prob=TRUE
# It takes some time, but the final table is reliable
## not run
# res <- contord(margins, Sigma, df=df, integerdf=FALSE, prob=TRUE)
# res$pij
#sum(res$pij) # OK!
## the probability P(X_1=1, X_2=3, X_3=2) is
# res$pij[1,3,2]
# Example 1
# consider a bivariate discrete random vector
k <- 2
# with these cumulative margins
marginal <- list(c(1/3, 2/3), c(0.1, 0.3, 0.6))
# generated by discretizing a multivariate standard normal variable
# with correlation matrix
Sigma <- matrix(c(1, .75, .75, 1), 2, 2)
# the resulting joint distribution and correlation matrix
# for the bivariate discrete random vector are
res <- contord(marginal, Sigma, prob=TRUE)
res$pij
res$SigmaO
# let's check if the margins are those assigned
cumsum(margin.table(res$pij,1))
cumsum(margin.table(res$pij,2))
# -> OK
# Example 2
# consider 4 discrete variables
k <- 4
# with these marginal distributions
marginal <- list(0.4,c(0.3,0.6), c(0.25,0.5,0.75), c(0.1,0.2,0.8,0.9))
# generated by discretizing a multivariate standard normal variable
# with correlation matrix
Sigma <- matrix(0.5,4,4)
diag(Sigma) <- 1
# the resulting correlation matrix for the discrete variables is
contord(marginal, Sigma)
# note that all the correlations are smaller than the original 0.5
# change Sigma, adding a negative correlation
Sigma[1,2] <- -0.15
Sigma[2,1] <- Sigma[1,2]
Sigma
# checking whether Sigma is still positive definite
eigen(Sigma)$values # all >0, OK
contord(marginal, Sigma)
# Example 2-bis
# the same margins and the same correlation matrix
# but now we consider a 4-variate Students's t with df=3
contord(marginal, Sigma, df=3)
# -> a slight reduction in magnitude for all the correlations
# Example 3
margin1 <- c(0.2,0.4,0.6,0.8)
margin2 <- margin1
marginal <- list(margin1, margin2)
sigma <- matrix(c(1,0.3,0.3,1),2,2)
df <- 3.5
contord(marginal=marginal, Sigma=sigma, df=df, integerdf=FALSE, prob=TRUE)
# compare with
contord(marginal=marginal, Sigma=sigma, df=round(df), prob=TRUE)
# Example 3
# multivariate probability tables
# Consider a t-copula-based model with these parameters:
margins <- list(1/2, c(1/3,2/3), c(1/6,1/2,5/6))
Sigma <- matrix(0.3, 3 ,3)
diag(Sigma) <- 1
df <- 3.5
# The 3-variate probability table of the discretized distribution is obtained by calling contord
# by setting integerdf=FALSE and prob=TRUE
# It takes some time, but the final table is reliable
## not run
# res <- contord(margins, Sigma, df=df, integerdf=FALSE, prob=TRUE)
# res$pij
#sum(res$pij) # OK!
## the probability P(X_1=1, X_2=3, X_3=2) is
# res$pij[1,3,2]

Checking correlations for feasibility

Description

The function returns the lower and upper bounds of the pairwise correlation coefficients of the discrete variables given their marginal distributions, i.e., returns the range of feasible bivariate correlations.

Usage

corrcheck(marginal, support = list(), Spearman = FALSE)
corrcheck(marginal, support = list(), Spearman = FALSE)

Arguments

marginal

support

Spearman

TRUE if we consider Spearman's correlation, FALSE (default) if we consider Pearson's correlation

Value

The function returns a list of two matrices: the former contains the lower bounds, the latter contains the upper bounds of the feasible pairwise correlations (on the extra-diagonal elements)

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

# four variables
k <- 4
# with 2, 3, 4, and 5 categories (Likert scales, by default)
kj <- 2:5
# and these marginal distributions (set of cumulative probabilities)
marginal <- list(0.4, c(0.6,0.9), c(0.1,0.2,0.4), c(0.6,0.7,0.8,0.9))
corrcheck(marginal) # lower and upper bounds for Pearson's rho
corrcheck(marginal, Spearman=TRUE) # lower and upper bounds for Spearman's rho
# change the supports
support <- list(c(0,1), c(1,2,4), c(1,2,3,4), c(0,1,2,5,10))
corrcheck(marginal, support=support) # updated bounds
# four variables
k <- 4
# with 2, 3, 4, and 5 categories (Likert scales, by default)
kj <- 2:5
# and these marginal distributions (set of cumulative probabilities)
marginal <- list(0.4, c(0.6,0.9), c(0.1,0.2,0.4), c(0.6,0.7,0.8,0.9))
corrcheck(marginal) # lower and upper bounds for Pearson's rho
corrcheck(marginal, Spearman=TRUE) # lower and upper bounds for Spearman's rho
# change the supports
support <- list(c(0,1), c(1,2,4), c(1,2,3,4), c(0,1,2,5,10))
corrcheck(marginal, support=support) # updated bounds

Estimation based on a bivariate sample of the multivariate discrete model obtained by discretizing a latent standard normal or Student's t distribution

Description

Limited to the bivariate case, based on an iid sample, the function estimates the parameters of the t-copula-based discrete distribution, according to either a two-step approach (suggested) where the only unknown parameters are the copula correlation and the degrees of freedom, whereas the two marginal probability distributions are estimated via the ecdf; or a full-maximum-likelihood approach, where also the two sets of marginal probabilities are estimated jointly with the copula correlation and degrees of freedom

Usage

estcontord(x, method="2-step", start.df=5, control= list(), normal=FALSE)
estcontord(x, method="2-step", start.df=5, control= list(), normal=FALSE)

Arguments

x

a matrix with two columns containing integer values; it is the bivariate iid sample

method

"2-step" for estimating the Student's parameters rho and df parametrically, after the margins have been estimated via the univariate ecdf; otherwise, a full-maximum-likelihood approach is carried out, where the marginal probabilities of the two random components are estimated as well

start.df

starting value for the degrees-of-freedom parameter (by default, set equal to 5)

control

a list of control options for the optim function in the maximum likelihood optimization routine

normal

a logical value; if FALSE, we estimate a discretized latent Student's t distribution; if TRUE, we estimate a discretized latent normal distribution

Value

a list containing the estimates (and for the "2-step" method their standard errors) along with the maximum log-likelihood value

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

# EX.1
x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
res <- estcontord(x)
res
# EX.2
set.seed(12345)
rho <- 0.5
df <- 5
Sigma <- matrix(c(1,rho,rho,1), 2, 2)
margins <- list(c(1/3,2/3), c(1/6,1/2,5/6))
# drawing a sample of size n=500
# from a discretized bivariate t distribution with rho=0.5 and df=5
# with the assigned margins above
x <- ordsample(n=500, marginal=margins, Sigma=Sigma, cormat="continuous", df=df)
# not run
#estcontord(x, "two-step")
## rho.hat ~ 0.477, df.hat ~ 7.02
#estcontord(x, "full", control=list(maxit=2000, factr = 1e6))
## rho.hat ~ 0.477, df.hat ~ 6.87
# EX.1
x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
res <- estcontord(x)
res
# EX.2
set.seed(12345)
rho <- 0.5
df <- 5
Sigma <- matrix(c(1,rho,rho,1), 2, 2)
margins <- list(c(1/3,2/3), c(1/6,1/2,5/6))
# drawing a sample of size n=500
# from a discretized bivariate t distribution with rho=0.5 and df=5
# with the assigned margins above
x <- ordsample(n=500, marginal=margins, Sigma=Sigma, cormat="continuous", df=df)
# not run
#estcontord(x, "two-step")
## rho.hat ~ 0.477, df.hat ~ 7.02
#estcontord(x, "full", control=list(maxit=2000, factr = 1e6))
## rho.hat ~ 0.477, df.hat ~ 6.87

Log-likelihood value for the t-copula-based model

Description

Negative log-likelihood value for the t-copula–based model to be used for two-step estimation in the bivariate case. The marginal probabilities are set equal to the corresponding relative frequencies of the input data set.

Usage

logL(arg, x)
logL(arg, x)

Arguments

arg

a vector containing the values of rho and df

x

a matrix with two columns containing the bivariate discrete data

Value

the negative log-likelihood value

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
r <- cor(x1,x2)
x <- cbind(x1,x2)
logL(arg=c(r, 4), x=x)
# it is the same as
logL_mle2(rho=r, df=4, x=x)
x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
r <- cor(x1,x2)
x <- cbind(x1,x2)
logL(arg=c(r, 4), x=x)
# it is the same as
logL_mle2(rho=r, df=4, x=x)

Log-likelihood function for the t-copula-based model

Description

Negative log-likelihood value for the t-copula-based model to be used for the two-step estimation in the bivariate case. The marginal probabilities are set equal to the corresponding relative frequencies of the input data set.

Usage

logL_mle2(rho, df, x)
logL_mle2(rho, df, x)

Arguments

rho

the value of rho

df

the value of df

x

a matrix with two columns containing the bivariate discrete data

Value

negativelog-likelihood value

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
logL_mle2(rho=0.5, df=5, x=x)
# it is the same as
logL(arg=c(0.5, 5), x=x)
x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
logL_mle2(rho=0.5, df=5, x=x)
# it is the same as
logL(arg=c(0.5, 5), x=x)

Log-likelihood value for the t-copula-based model

Description

Negative log-likelihood value for the t-copula-based model, to be used for estimation in the bivariate case

Usage

logLfull(arg, x)
logLfull(arg, x)

Arguments

arg

a vector containing the values of the correlation parameter rho, of the $m_1$ marginal probabilities for rv $X_1$ , of the $m_2$ marginal probabilities for rv $X_2$ , and, finally, of the degrees-of-freedom parameter df

x

a matrix with two columns containing the bivariate discrete data

Details

The vector arg contains in position $1$ the correlation parameter rho, from position $2$ to position $m_1+1$ the $m_1$ values of the marginal probabilities of $X_1$ ; from position $m_1+2$ to position $m_1+m_2+1$ the $m_2$ values of the marginal probabilities of $X_2$ . The position $m_1+m_2+2$ contains, if applicable, the degrees-of-freedom parameter.

Value

the negative value of the log-likelihood function

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
logLfull(arg=c(0.5,rep(c(0.44, 0.54, 0.02),2),5), x=x)
x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
logLfull(arg=c(0.5,rep(c(0.44, 0.54, 0.02),2),5), x=x)

Log-likelihood value for the t-copula-based model (with marginal probabilities transformed into unconstrained variables)

Description

Negative log-likelihood value for the t-copula-based model to be used for estimation in the bivariate case

Usage

logLfullz(arg, x)
logLfullz(arg, x)

Arguments

arg

a vector containing the values of the correlation parameter rho, of the softmax-transformed $m_1$ marginal probabilities for rv $X_1$ , of the softmax-transformed $m_2$ marginal probabilities for rv $X_2$ , and, finally, of the degrees-of-freedom parameter df

x

a matrix with two columns containing the bivariate discrete data

Details

The vector arg contains in position $1$ the correlation parameter rho, from position $2$ to position $m_1+1$ the $m_1$ values $z_i$ from which the marginal probabilities of $X_1$ are obtained as $p_{1i}=\exp(z_i)/\sum_{i=1}^{m_1} exp(z_i)$ ; from position $m_1+2$ to position $m_1+m_2+1$ the $m_2$ values $w_j$ from which the marginal probabilities of $X_2$ are obtained as $p_{2j}=\exp(w_j)/\sum_{j=1}^{m_2} exp(w_j)$ . The position $m_1+m_2+2$ contains, if applicable, the degrees-of-freedom parameter.

Value

the negative value of the log-likelihood function

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
logLfullz(arg=c(0.5,rep(c(-0.5, 0, -3),2),5), x=x)
x1 <- c(rep(0,223),rep(1,269),rep(2,8))
x2 <- c(rep(0,153), rep(1,70), rep(0,75),rep(1,187),
        rep(2,7),rep(0,2),rep(1,4),rep(2,2))
cor(x1,x2)
x <- cbind(x1,x2)
logLfullz(arg=c(0.5,rep(c(-0.5, 0, -3),2),5), x=x)

Computing the "intermediate" correlation matrix for the multivariate standard normal or Student's t in order to achieve the "target" correlation matrix for the multivariate discrete variable

Description

The function computes the correlation matrix of the $k$ -dimensional standard normal or Student's t random variable that yields the desired correlation matrix Sigma for the $k$ -dimensional r.v. with desired marginal distributions

Usage

ordcont(marginal, Sigma, support = list(), Spearman = FALSE,
epsilon = 1e-06, maxit = 100, df=Inf)
ordcont(marginal, Sigma, support = list(), Spearman = FALSE,
epsilon = 1e-06, maxit = 100, df=Inf)

Arguments

marginal

Sigma

the target correlation matrix of the discrete variables

support

Spearman

if TRUE, the function finds Spearman's correlations (and it is not necessary to provide support), if FALSE (default) Pearson's correlations

epsilon

the maximum tolerated error between target and actual correlations

maxit

the maximum number of iterations allowed for the algorithm

df

the degrees of freedom of the multivariate Student's t. By default, set equal to Inf, corresponding to the multivariate normal

Details

This function implements what is sometimes called the “inverse problem” or the “matching correlation problem” (see the references), that is, it seeks the correlation matrix of the (latent) continuous (normal or Student's t) random vector that ensures a target correlation matrix for the (observed) discrete random vector, given the margins for the latter (and the number of degrees of freedom for the former, if Student's t)

Value

a list of five elements

SigmaC

the correlation matrix of the multivariate standard normal variable

SigmaO

the actual correlation matrix of the discretized variables (it should approximately coincide with the target correlation matrix Sigma)

Sigma

the target correlation matrix of the discrete variables

niter

a matrix containing the number of iterations performed by the algorithm, one for each pair of variables

maxerr

the actual maximum error (the maximum absolute deviation between actual and target correlations of the discrete variables)

Note

For some choices of marginal and Sigma, there may not exist a feasible $k$ -variate probability mass function or the algorithm may not provide a feasible correlation matrix SigmaC (see, for example, Changanty and Joe, 2006). In this case, the procedure stops and exits with an error. The value of the maximum tolerated absolute error epsilon on the elements of the correlation matrix for the target r.v. can be set by the user: a value between 1e-6 and 1e-2 appears to be an acceptable compromise ensuring both precision of the results and convergence of the algorithm; moreover, a maximum number of iterations can be chosen (maxit), in order to avoid possible endless loops

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

References

Ferrari, P.A., Barbiero, A.: Simulating ordinal data. Multivariate Behavioral Research 47(4), 1–24 (2012)

Cario, M., Nelson, B.: Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois (1997)

Chaganty, N. R., Joe, H. (2006). Range of correlation matrices for dependent Bernoulli random variables. Biometrika, 93(1), 197-206.

Xiao, Q.: Generating correlated random vector involving discrete variables. Communications in Statistics - Theory and Methods 46(4), 1594–1605 (2017)

Examples

# EX.1
# consider a 4-dimensional ordinal variable
k <- 4
# with different number of categories
kj <- 2:5
# and uniform marginal distributions with variable number of categories
marginal <- list(0.5, (1:2)/3, (1:3)/4, (1:4)/5)
corrcheck(marginal)
# and the following correlation matrix
Sigma <- matrix(c(1,0.5,0.4,0.3,0.5,1,0.5,0.4,0.4,0.5,1,0.5,0.3,0.4,0.5,1),
4, 4, byrow=TRUE)
Sigma
# the correlation matrix of the standard 4-dimensional standard normal or Student's t
# ensuring Sigma is
res <- ordcont(marginal, Sigma)
res[[1]]
# change some marginal distributions
marginal <- list(0.3, c(1/3, 2/3), c(1/5, 2/5, 3/5), c(0.1, 0.2, 0.4, 0.6))
corrcheck(marginal)
# and notice how the correlation matrix of the multivariate normal changes...
res <- ordcont(marginal, Sigma)
res[[1]]
# change Sigma, adding a negative correlation
Sigma[1,2] <- -0.2
Sigma[2,1] <- Sigma[1,2]
Sigma
# checking whether Sigma is still positive definite
eigen(Sigma)$values # all >0, OK
res <- ordcont(marginal, Sigma)
res[[1]]
# consider now a multivariate Student's t with 10 dof
res.t <- ordcont(marginal, Sigma, df=10)
res.t$SigmaC
res.t$SigmaO
# EX.2
# contord and ordcont
margins <- list(1/2, c(1/3,2/3), c(1/6,1/2,5/6))
Sigma <- matrix(0.3, 3 ,3)
diag(Sigma) <- 1
Sigma
df <- 3
# the "direct problem"
res.co <- contord(margins, Sigma, df=df)
res.co
# the "inverse problem"
res.oc <- ordcont(margins, Sigma=res.co, df=df)
res.oc$SigmaC # it is almost equal to Sigma
# EX.1
# consider a 4-dimensional ordinal variable
k <- 4
# with different number of categories
kj <- 2:5
# and uniform marginal distributions with variable number of categories
marginal <- list(0.5, (1:2)/3, (1:3)/4, (1:4)/5)
corrcheck(marginal)
# and the following correlation matrix
Sigma <- matrix(c(1,0.5,0.4,0.3,0.5,1,0.5,0.4,0.4,0.5,1,0.5,0.3,0.4,0.5,1),
4, 4, byrow=TRUE)
Sigma
# the correlation matrix of the standard 4-dimensional standard normal or Student's t
# ensuring Sigma is
res <- ordcont(marginal, Sigma)
res[[1]]
# change some marginal distributions
marginal <- list(0.3, c(1/3, 2/3), c(1/5, 2/5, 3/5), c(0.1, 0.2, 0.4, 0.6))
corrcheck(marginal)
# and notice how the correlation matrix of the multivariate normal changes...
res <- ordcont(marginal, Sigma)
res[[1]]
# change Sigma, adding a negative correlation
Sigma[1,2] <- -0.2
Sigma[2,1] <- Sigma[1,2]
Sigma
# checking whether Sigma is still positive definite
eigen(Sigma)$values # all >0, OK
res <- ordcont(marginal, Sigma)
res[[1]]
# consider now a multivariate Student's t with 10 dof
res.t <- ordcont(marginal, Sigma, df=10)
res.t$SigmaC
res.t$SigmaO
# EX.2
# contord and ordcont
margins <- list(1/2, c(1/3,2/3), c(1/6,1/2,5/6))
Sigma <- matrix(0.3, 3 ,3)
diag(Sigma) <- 1
Sigma
df <- 3
# the "direct problem"
res.co <- contord(margins, Sigma, df=df)
res.co
# the "inverse problem"
res.oc <- ordcont(margins, Sigma=res.co, df=df)
res.oc$SigmaC # it is almost equal to Sigma

Drawing a sample of discrete data

Description

The function draws a sample from a multivariate discrete variable obtained by discretizing a multivariate Student's t or Gaussian random variable with pressribed correlation matrix Sigma and marginal distributions marginal

Usage

ordsample(n, marginal, Sigma, support = list(), Spearman = FALSE,
cormat = "discrete", df=Inf)
ordsample(n, marginal, Sigma, support = list(), Spearman = FALSE,
cormat = "discrete", df=Inf)

Arguments

n

the sample size

marginal

a list of $k$ elements, where $k$ is the number of variables. The $i$ -th element of marginal is the vector of the cumulative probabilities defining the marginal distribution of the $i$ -th component of the multivariate variable. If the $i$ -th component can take $k_i$ values, the $i$ -th element of marginal will contain $k_i-1$ probabilities (the $k_i$ -th is obviously 1 and shall not be included).

Sigma

the target correlation matrix of the multivariate latent t or Gaussian random variable or of the multivariate discrete random variable

support

Spearman

if TRUE, the function finds Spearman's correlations (and it is not necessary to provide support), if FALSE (default) Pearson's correlations

cormat

"discrete" if the Sigma in input is the target correlation matrix of the multivariate discrete variable; "continuous" if the Sigma in input is the intermediate correlation matrix of the multivariate t or Gaussian random variable

df

the degrees of freedom of the multivariate Student's t distribution. By default, set equal to Inf, corresponding to the multivariate normal

Value

A $n\times k$ matrix of values drawn from the $k$ -variate discrete r.v. with the desired marginal distributions and correlation matrix

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

# Example 1

# draw a sample from a bivariate ordinal variable
# with 4 categories and asymmetrical marginal distributions
# and correlation coefficient 0.6 (to be checked)
k <- 2
marginal <- list(c(0.1,0.3,0.6), c(0.4,0.7,0.9))
corrcheck(marginal) # check ok
Sigma <- matrix(c(1,0.6,0.6,1),2,2)
# sample size 1000
n <- 1000
# generate a sample of size n
m <- ordsample(n, marginal, Sigma)
head(m)
# sample correlation matrix
cor(m) # compare it to Sigma
# empirical marginal distributions
cumsum(table(m[,1]))/n
cumsum(table(m[,2]))/n # compare them with the two marginal distributions

# Example 1bis

# draw a sample from a bivariate ordinal variable
# with 4 categories and asymmetrical marginal distributions
# and Spearman correlation coefficient 0.6 (to be checked)
k <- 2
marginal <- list(c(0.1,0.3,0.6), c(0.4,0.7,0.9))
corrcheck(marginal, Spearman=TRUE) # check ok
Sigma <- matrix(c(1,0.6,0.6,1), 2, 2)
# sample size 1000
n <- 1000
# generate a sample of size n
m <- ordsample(n, marginal, Sigma, Spearman=TRUE)
head(m)
# sample correlation matrix
cor(rank(m[,1]),rank(m[,2])) # compare it with Sigma
# empirical marginal distributions
cumsum(table(m[,1]))/n
cumsum(table(m[,2]))/n # compare them with the two marginal distributions

# Example 1ter

# draw a sample from a bivariate random variable
# with binomial marginal distributions (n=3, p=1/3 and n=4, p=2/3)
# and Pearson correlation coefficient 0.6 (to be checked)
k <- 2
marginal <- list(pbinom(0:2, 3, 1/3),pbinom(0:3, 4, 2/3))
marginal
corrcheck(marginal, support=list(0:3, 0:4)) # check ok
Sigma <- matrix(c(1,0.6,0.6,1), 2, 2)
# sample size 1000
n <- 1000
# generate a sample of size n
m <- ordsample(n, marginal, Sigma, support=list(0:3,0:4))
head(m)
# sample correlation matrix
cor(m) # compare it with Sigma
# empirical marginal distributions
cumsum(table(m[,1]))/n
cumsum(table(m[,2]))/n # compare them with the two marginal distributions

# Example 2

# draw a sample from a 4-dimensional ordinal variable
# with different number of categories and uniform marginal distributions
# and different correlation coefficients
k <- 4
marginal <- list(0.5, c(1/3,2/3), c(1/4,2/4,3/4), c(1/5,2/5,3/5,4/5))
corrcheck(marginal)
# select a feasible correlation matrix
Sigma <- matrix(c(1,0.5,0.4,0.3,0.5,1,0.5,0.4,0.4,0.5,1,0.5,0.3,0.4,0.5,1),
4, 4, byrow=TRUE)
Sigma
# sample size 100
n <- 100
# generate a sample of size n
set.seed(1)
m <- ordsample(n, marginal, Sigma)
# sample correlation matrix
cor(m) # compare it with Sigma
# empirical marginal distribution
cumsum(table(m[,4]))/n # compare it with the fourth marginal
head(m)
# or equivalently...
set.seed(1)
res <- ordcont(marginal, Sigma)
res[[1]] # the intermediate correlation matrix of the multivariate normal
m <- ordsample(n, marginal, res[[1]], cormat="continuous")
head(m)

# Example 3
# simulation of two correlated Poisson r.v.
# modification to GenOrd sampling function for Poisson distribution
ordsamplep <- function (n, lambda, Sigma) 
{
   k <- length(lambda)
   valori <- mvtnorm::rmvnorm(n, rep(0, k), Sigma)
   for (i in 1:k)
   {
   valori[, i] <- qpois(pnorm(valori[,i]), lambda[i])
   }
   return(valori)
}
# number of variables
k <- 2
# Poisson parameters
lambda <- c(2, 5)
# correlation matrix
Sigma <- matrix(0.25, 2, 2)
diag(Sigma) <- 1
# sample size
n <- 10000
# preliminary stage: support TRUNCATION
# required for recovering the correlation matrix
# of the standard bivariate normal
# truncation error
epsilon <- 0.0001
# corresponding maximum value
kmax <- qpois(1-epsilon, lambda)
# truncated marginals
l <- list()
for(i in 1:k)
{
l[[i]] <- 0:kmax[i]
}
marg <- list()
for(i in 1:k)
{
marg[[i]] <- dpois(0:kmax[i],lambda[i])
marg[[i]][kmax[i]+1] <- 1-sum(marg[[i]][1:(kmax[i])])
}
cm <- list()
for(i in 1:k)
{
cm[[i]] <- cumsum(marg[[i]])
cm[[i]] <- cm[[i]][-(kmax[i]+1)]
}
# check feasibility of correlation matrix
RB <- corrcheck(cm, support=l)
RL <- RB[[1]]
RU <- RB[[2]]
Sigma <= RU & Sigma >= RL # OK
res <- ordcont(cm, Sigma, support=l)
res[[1]]
Sigma <- res[[1]]
# draw the sample
m <- ordsamplep(n, lambda, Sigma)
# sample correlation matrix
cor(m)
head(m)

# Example 4
# simulation of 4 correlated binary and Poisson r.v.'s (2+2)
# modification to GenOrd sampling function
ordsamplep <- function (n, marginal, lambda, Sigma) 
{
k <- length(lambda)
valori <- mvtnorm::rmvnorm(n, rep(0, k), Sigma)
for(i in 1:k)
{
if(lambda[i]==0)
{
	valori[, i] <- as.integer(cut(valori[, i], breaks = c(min(valori[,i]) - 1,
               qnorm(marginal[[i]]), max(valori[, i]) + 1)))
valori[, i] <- support[[i]][valori[, i]]
}
else
{
	valori[, i] <- qpois(pnorm(valori[,i]), lambda[i])
}
	}
return(valori)
}
# number of variables
k <- 4
# Poisson parameters (only 3rd and 4th are Poisson)
lambda <- c(0, 0, 2, 5)
# 1st and 2nd are Bernoulli with p=0.5
marginal <- list()
marginal[[1]] <- .5
marginal[[2]] <- .5
marginal[[3]] <- 0
marginal[[4]] <- 0
# support
support <- list()
support[[1]] <- 0:1
support[[2]] <- 0:1
# correlation matrix
Sigma <- matrix(0.25, k, k)
diag(Sigma) <- 1
# sample size
n <- 10000
# preliminary stage: support TRUNCATION
# required for recovering the correlation matrix
# of the standard bivariate normal
# truncation error
epsilon <- 0.0001
# corresponding maximum value
kmax <- qpois(1-epsilon, lambda)
# truncated marginals
for(i in 3:4)
{
support[[i]] <- 0:kmax[i]
}
marg <- list()
for(i in 3:4)
{
marg[[i]] <- dpois(0:kmax[i],lambda[i])
marg[[i]][kmax[i]+1] <- 1-sum(marg[[i]][1:(kmax[i])])
}
for(i in 3:4)
{
marginal[[i]] <- cumsum(marg[[i]])
marginal[[i]] <- marginal[[i]][-(kmax[i]+1)]
}
# check feasibility of correlation matrix
RB <- corrcheck(marginal, support=support)
RL <- RB[[1]]
RU <- RB[[2]]
Sigma <= RU & Sigma >= RL # OK
# compute the correlation matrix of the 4-variate standard normal
res <- ordcont(marginal, Sigma, support=support)
res[[1]]
Sigma <- res[[1]]
# draw the sample
m <- ordsamplep(n, marginal, lambda, Sigma)
# sample correlation matrix
cor(m)
head(m)
# Example 1

# draw a sample from a bivariate ordinal variable
# with 4 categories and asymmetrical marginal distributions
# and correlation coefficient 0.6 (to be checked)
k <- 2
marginal <- list(c(0.1,0.3,0.6), c(0.4,0.7,0.9))
corrcheck(marginal) # check ok
Sigma <- matrix(c(1,0.6,0.6,1),2,2)
# sample size 1000
n <- 1000
# generate a sample of size n
m <- ordsample(n, marginal, Sigma)
head(m)
# sample correlation matrix
cor(m) # compare it to Sigma
# empirical marginal distributions
cumsum(table(m[,1]))/n
cumsum(table(m[,2]))/n # compare them with the two marginal distributions

# Example 1bis

# draw a sample from a bivariate ordinal variable
# with 4 categories and asymmetrical marginal distributions
# and Spearman correlation coefficient 0.6 (to be checked)
k <- 2
marginal <- list(c(0.1,0.3,0.6), c(0.4,0.7,0.9))
corrcheck(marginal, Spearman=TRUE) # check ok
Sigma <- matrix(c(1,0.6,0.6,1), 2, 2)
# sample size 1000
n <- 1000
# generate a sample of size n
m <- ordsample(n, marginal, Sigma, Spearman=TRUE)
head(m)
# sample correlation matrix
cor(rank(m[,1]),rank(m[,2])) # compare it with Sigma
# empirical marginal distributions
cumsum(table(m[,1]))/n
cumsum(table(m[,2]))/n # compare them with the two marginal distributions

# Example 1ter

# draw a sample from a bivariate random variable
# with binomial marginal distributions (n=3, p=1/3 and n=4, p=2/3)
# and Pearson correlation coefficient 0.6 (to be checked)
k <- 2
marginal <- list(pbinom(0:2, 3, 1/3),pbinom(0:3, 4, 2/3))
marginal
corrcheck(marginal, support=list(0:3, 0:4)) # check ok
Sigma <- matrix(c(1,0.6,0.6,1), 2, 2)
# sample size 1000
n <- 1000
# generate a sample of size n
m <- ordsample(n, marginal, Sigma, support=list(0:3,0:4))
head(m)
# sample correlation matrix
cor(m) # compare it with Sigma
# empirical marginal distributions
cumsum(table(m[,1]))/n
cumsum(table(m[,2]))/n # compare them with the two marginal distributions

# Example 2

# draw a sample from a 4-dimensional ordinal variable
# with different number of categories and uniform marginal distributions
# and different correlation coefficients
k <- 4
marginal <- list(0.5, c(1/3,2/3), c(1/4,2/4,3/4), c(1/5,2/5,3/5,4/5))
corrcheck(marginal)
# select a feasible correlation matrix
Sigma <- matrix(c(1,0.5,0.4,0.3,0.5,1,0.5,0.4,0.4,0.5,1,0.5,0.3,0.4,0.5,1),
4, 4, byrow=TRUE)
Sigma
# sample size 100
n <- 100
# generate a sample of size n
set.seed(1)
m <- ordsample(n, marginal, Sigma)
# sample correlation matrix
cor(m) # compare it with Sigma
# empirical marginal distribution
cumsum(table(m[,4]))/n # compare it with the fourth marginal
head(m)
# or equivalently...
set.seed(1)
res <- ordcont(marginal, Sigma)
res[[1]] # the intermediate correlation matrix of the multivariate normal
m <- ordsample(n, marginal, res[[1]], cormat="continuous")
head(m)

# Example 3
# simulation of two correlated Poisson r.v.
# modification to GenOrd sampling function for Poisson distribution
ordsamplep <- function (n, lambda, Sigma) 
{
   k <- length(lambda)
   valori <- mvtnorm::rmvnorm(n, rep(0, k), Sigma)
   for (i in 1:k)
   {
   valori[, i] <- qpois(pnorm(valori[,i]), lambda[i])
   }
   return(valori)
}
# number of variables
k <- 2
# Poisson parameters
lambda <- c(2, 5)
# correlation matrix
Sigma <- matrix(0.25, 2, 2)
diag(Sigma) <- 1
# sample size
n <- 10000
# preliminary stage: support TRUNCATION
# required for recovering the correlation matrix
# of the standard bivariate normal
# truncation error
epsilon <- 0.0001
# corresponding maximum value
kmax <- qpois(1-epsilon, lambda)
# truncated marginals
l <- list()
for(i in 1:k)
{
l[[i]] <- 0:kmax[i]
}
marg <- list()
for(i in 1:k)
{
marg[[i]] <- dpois(0:kmax[i],lambda[i])
marg[[i]][kmax[i]+1] <- 1-sum(marg[[i]][1:(kmax[i])])
}
cm <- list()
for(i in 1:k)
{
cm[[i]] <- cumsum(marg[[i]])
cm[[i]] <- cm[[i]][-(kmax[i]+1)]
}
# check feasibility of correlation matrix
RB <- corrcheck(cm, support=l)
RL <- RB[[1]]
RU <- RB[[2]]
Sigma <= RU & Sigma >= RL # OK
res <- ordcont(cm, Sigma, support=l)
res[[1]]
Sigma <- res[[1]]
# draw the sample
m <- ordsamplep(n, lambda, Sigma)
# sample correlation matrix
cor(m)
head(m)

# Example 4
# simulation of 4 correlated binary and Poisson r.v.'s (2+2)
# modification to GenOrd sampling function
ordsamplep <- function (n, marginal, lambda, Sigma) 
{
k <- length(lambda)
valori <- mvtnorm::rmvnorm(n, rep(0, k), Sigma)
for(i in 1:k)
{
if(lambda[i]==0)
{
	valori[, i] <- as.integer(cut(valori[, i], breaks = c(min(valori[,i]) - 1,
               qnorm(marginal[[i]]), max(valori[, i]) + 1)))
valori[, i] <- support[[i]][valori[, i]]
}
else
{
	valori[, i] <- qpois(pnorm(valori[,i]), lambda[i])
}
	}
return(valori)
}
# number of variables
k <- 4
# Poisson parameters (only 3rd and 4th are Poisson)
lambda <- c(0, 0, 2, 5)
# 1st and 2nd are Bernoulli with p=0.5
marginal <- list()
marginal[[1]] <- .5
marginal[[2]] <- .5
marginal[[3]] <- 0
marginal[[4]] <- 0
# support
support <- list()
support[[1]] <- 0:1
support[[2]] <- 0:1
# correlation matrix
Sigma <- matrix(0.25, k, k)
diag(Sigma) <- 1
# sample size
n <- 10000
# preliminary stage: support TRUNCATION
# required for recovering the correlation matrix
# of the standard bivariate normal
# truncation error
epsilon <- 0.0001
# corresponding maximum value
kmax <- qpois(1-epsilon, lambda)
# truncated marginals
for(i in 3:4)
{
support[[i]] <- 0:kmax[i]
}
marg <- list()
for(i in 3:4)
{
marg[[i]] <- dpois(0:kmax[i],lambda[i])
marg[[i]][kmax[i]+1] <- 1-sum(marg[[i]][1:(kmax[i])])
}
for(i in 3:4)
{
marginal[[i]] <- cumsum(marg[[i]])
marginal[[i]] <- marginal[[i]][-(kmax[i]+1)]
}
# check feasibility of correlation matrix
RB <- corrcheck(marginal, support=support)
RL <- RB[[1]]
RU <- RB[[2]]
Sigma <= RU & Sigma >= RL # OK
# compute the correlation matrix of the 4-variate standard normal
res <- ordcont(marginal, Sigma, support=support)
res[[1]]
Sigma <- res[[1]]
# draw the sample
m <- ordsamplep(n, marginal, lambda, Sigma)
# sample correlation matrix
cor(m)
head(m)

Rectangle Probability for a Bivariate Student's t Distribution

Description

Computes the probability that a bivariate Student's t random vector with correlation parameter rho and df degrees of freedom (non necessarily integer) falls in a rectangular region.

Usage

p_rect_t(ax, bx, ay, by, rho, df, rel.tol = 1e-4)
p_rect_t(ax, bx, ay, by, rho, df, rel.tol = 1e-4)

Arguments

ax

lower bound for the first component

bx

upper bound for the first component

ay

lower bound for the second component

by

upper bound for the second component

rho

correlation parameter of the bivariate Student's t distribution.

df

degrees-of-freedom parameter

rel.tol

relative accuracy requested for numerical integration; see integrate.

Details

The function evaluates

$P(a_x < X \le b_x, a_y < Y \le b_y),$

where $(X,Y)$ follows a standard bivariate Student's t distribution with correlation parameter rho and df degrees of freedom.

The probability is computed through one-dimensional numerical integration after transformation to the marginal t probability scale. Infinite lower and upper ounds are allowed.

Value

A numeric value giving the rectangle probability.

Author(s)

Alessandro Barbiero

Examples

## Probability over a finite rectangle
p_rect_t(ax = -1, bx = 0, ay = -0.5, by = 1, rho = 0.5, df = 3.5)

## Probability with an infinite lower bound
p_rect_t(ax = -Inf, bx = 0, ay = -0.5, by = 1, rho = 0.5, df = 5)
## Probability over a finite rectangle
p_rect_t(ax = -1, bx = 0, ay = -0.5, by = 1, rho = 0.5, df = 3.5)

## Probability with an infinite lower bound
p_rect_t(ax = -Inf, bx = 0, ay = -0.5, by = 1, rho = 0.5, df = 5)

Probabilities for a multivariate t with non-integer degrees-of-freedom parameter

Description

Computes probabilities of an hyper-rectangle for a standard multivariate Student's $t$ with non-integer degrees-of-freedom parameter, directly integrating the function dmvt, available in the package mvtnorm, through the cuhre function, available in the package cubature

Usage

pmvt.alt(low, upp, corr, df)
pmvt.alt(low, upp, corr, df)

Arguments

low

a vector containing the lower bounds

upp

a vector containing the upper bounds

corr

the correlation matrix of the $t$ copula

df

the degrees-of-freedom parameter, possibly non-integer

Author(s)

Alessandro Barbiero, Pier Alda Ferrari

Examples

# dimension 2
lower <- c(-1, -1/2)
upper <- c(0, 1)
Sigma <- matrix(c(1, 0.5, 0.5, 1), 2 , 2)
pmvt.alt(low=lower, upp=upper, corr=Sigma, df=3.5)
# dimension 4
lower <- c(-1, -1/2, 1/2, -1/4)
upper <- c(0, 1, 2, 3/4)
Sigma <- matrix(0.25, 4, 4)
diag(Sigma) <- 1
pmvt.alt(low=lower, upp=upper, corr=Sigma, df=3.5)
# let's set df=Inf
pmvt.alt(low=lower, upp=upper, corr=Sigma, df=Inf)
# this probability should be equal to
mvtnorm::pmvnorm(lower=lower, upper=upper, corr=Sigma) # OK!
# dimension 2
lower <- c(-1, -1/2)
upper <- c(0, 1)
Sigma <- matrix(c(1, 0.5, 0.5, 1), 2 , 2)
pmvt.alt(low=lower, upp=upper, corr=Sigma, df=3.5)
# dimension 4
lower <- c(-1, -1/2, 1/2, -1/4)
upper <- c(0, 1, 2, 3/4)
Sigma <- matrix(0.25, 4, 4)
diag(Sigma) <- 1
pmvt.alt(low=lower, upp=upper, corr=Sigma, df=3.5)
# let's set df=Inf
pmvt.alt(low=lower, upp=upper, corr=Sigma, df=Inf)
# this probability should be equal to
mvtnorm::pmvnorm(lower=lower, upper=upper, corr=Sigma) # OK!

Package 'GenOrd'

Help Index

Simulation of Discrete Random Variables with a Given Marginal Distributions and Correlation Matrix

Description

Details

Author(s)

References

See Also

Correlations of discretized variables

Description

Usage

Arguments

Value

Note

Author(s)

See Also

Examples

Checking correlations for feasibility

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Estimation based on a bivariate sample of the multivariate discrete model obtained by discretizing a latent standard normal or Student's t distribution

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Log-likelihood value for the t-copula-based model

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Log-likelihood function for the t-copula-based model

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Log-likelihood value for the t-copula-based model

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Log-likelihood value for the t-copula-based model (with marginal probabilities transformed into unconstrained variables)

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Computing the "intermediate" correlation matrix for the multivariate standard normal or Student's t in order to achieve the "target" correlation matrix for the multivariate discrete variable

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Drawing a sample of discrete data

Description