Type:	Package
Title:	Moments of Folded and Doubly Truncated Multivariate Distributions
Version:	6.1
Date:	2024-10-17
Author:	Christian E. Galarza [aut, cre, trl], Raymond Kan [ctb], Victor H. Lachos [aut, ths]
Maintainer:	Christian E. Galarza <cgalarza88@gmail.com>
Description:	It computes arbitrary products moments (mean vector and variance-covariance matrix), for some double truncated (and folded) multivariate distributions. These distributions belong to the family of selection elliptical distributions, which includes well known skewed distributions as the unified skew-t distribution (SUT) and its particular cases as the extended skew-t (EST), skew-t (ST) and the symmetric student-t (T) distribution. Analogous normal cases unified skew-normal (SUN), extended skew-normal (ESN), skew-normal (SN), and symmetric normal (N) are also included. Density, probabilities and random deviates are also offered for these members.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Depends:	R (≥ 3.6.0)
Imports:	Rcpp (≥ 1.0.1), mvtnorm (≥ 1.0.11), tlrmvnmvt (≥ 1.1.0), hypergeo
LinkingTo:	Rcpp (≥ 1.0.1), RcppArmadillo, mvtnorm
Suggests:	tmvtnorm
NeedsCompilation:	yes
Packaged:	2024-10-28 21:14:51 UTC; cgala
Repository:	CRAN
Date/Publication:	2024-10-28 21:40:02 UTC

Moments of Folded and Doubly Truncated Multivariate Distributions

Description

It computes arbitrary products moments (mean vector and variance-covariance matrix), for some double truncated (and folded) multivariate distributions. These distributions belong to the family of selection elliptical distributions, which includes well known skewed distributions as the unified skew-t distribution (SUT) and its particular cases as the extended skew-t (EST), skew-t (ST) and the symmetric student-t (T) distribution. Analogous normal cases unified skew-normal (SUN), extended skew-normal (ESN), skew-normal (SN), and symmetric normal (N) are also included. Density, probabilities and random deviates are also offered for these members.

Details

Probabilities can be computed using the functions pmvSN and pmvESN for the normal cases SN and ESN and, pmvST and pmvEST for the t cases ST and EST respectively, which offer the option to return the logarithm in base 2 of the probability, useful when the true probability is too small for the machine precision. These functions above use methods in Genz (1992) through the mvtnorm package (linked direclty to our C++ functions) and Cao et.al. (2019) through the package tlrmvnmvt. For the double truncated Student-t cases SUT, EST, ST and T, decimal degrees of freedom are supported. Computation of arbitrary moments are based in the works of Kan & Robotti (2017) and Galarza et.al. (2021,2022a,2022b). Reference for the family of selection-elliptical distributions in this package can be found in Arellano-Valle & Genton (2005).

Author(s)

Christian E. Galarza [aut, cre, trl] (<https://orcid.org/0000-0002-4818-6006>), Raymond Kan [ctb] (<https://orcid.org/0000-0002-0578-9974>), Victor H. Lachos [aut, ths] (<https://orcid.org/0000-0002-7239-2459>)

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Arellano-Valle, R. B. & Genton, M. G. (2005). On fundamental skew distributions. Journal of Multivariate Analysis, 96, 93-116.

Cao, J., Genton, M. G., Keyes, D. E., & Turkiyyah, G. M. (2019) "Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student-t Probabilities" <https://marcgenton.github.io/2019.CGKT.manuscript.pdf>.

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Genz, A., "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149 (1992) <doi:10.1080/10618600.1992.10477010>.

Kan, R., & Robotti, C. (2017). On moments of folded and truncated multivariate normal distributions. Journal of Computational and Graphical Statistics, 26(4), 930-934.

See Also

onlymeanTMD,meanvarTMD,momentsTMD,dmvSN,pmvSN,rmvSN,dmvST,pmvST,rmvST

Examples

a = c(-0.8,-0.7,-0.6)
b = c(0.5,0.6,0.7)
mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)

meanvarTMD(a,b,mu,Sigma,dist="normal") #normal case
meanvarTMD(mu = mu,Sigma = Sigma,lambda = c(-2,0,1),dist="SN") #skew normal with NO truncation
meanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),nu = 4.87,dist = "ST") #skew t
momentsTMD(3,a,b,mu,Sigma,nu = 4,dist = "t") #t case, all moments or order <=3

Monte Carlo Mean and variance for doubly truncated multivariate distributions

Description

It computes the Monte Carlo mean vector and variance-covariance matrix for some doubly truncated skew-elliptical distributions. Monte Carlo simulations are performed via slice Sampling. It supports the p-variate Normal, Skew-normal (SN), Extended Skew-normal (ESN) and Unified Skew-normal (SUN) as well as the Student's-t, Skew-t (ST), Extended Skew-t (EST) and Unified Skew-t (SUT) distribution.

Usage

MCmeanvarTMD(lower = rep(-Inf,length(mu)),upper = rep(Inf,length(mu)),mu,Sigma
,lambda = NULL,tau = NULL,Gamma = NULL,nu = NULL,dist,n = 10000)

Arguments

Value

It returns a list with three elements:

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Arellano-Valle, R. B. & Genton, M. G. (2005). On fundamental skew distributions. Journal of Multivariate Analysis, 96, 93-116.

Ho, H. J., Lin, T. I., Chen, H. Y., & Wang, W. L. (2012). Some results on the truncated multivariate t distribution. Journal of Statistical Planning and Inference, 142(1), 25-40.

See Also

meanvarTMD, rmvSN,rmvESN,rmvST, rmvEST

Examples

a = c(-0.8,-0.7,-0.6)
b = c(0.5,0.6,0.7)
mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)


## Normal case

# Theoretical value
value1 = meanvarTMD(a,b,mu,Sigma,dist="normal")

#MC estimate
MC11 = MCmeanvarTMD(a,b,mu,Sigma,dist="normal") #by defalut n = 10000
MC12 = MCmeanvarTMD(a,b,mu,Sigma,dist="normal",n = 10^5) #more precision

## Skew-t case
 
# Theoretical value
value2 = meanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),nu = 4,dist = "ST")

#MC estimate
MC21 = MCmeanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),nu = 4,dist = "ST")

## More...

MC5 = MCmeanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")
MC6 = MCmeanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,nu = 4,dist = "EST")


#Skew-unified Normal (SUN) and Skew-unified t (SUT) distributions

Lambda = matrix(c(1,0,2,-3,0,-1),3,2) #A skewness matrix p times q
Gamma  = matrix(c(1,-0.5,-0.5,1),2,2) #A correlation matrix q times q
tau    = c(-1,2)                      #A vector of extension parameters of dim q

MC7 = MCmeanvarTMD(a,b,mu,Sigma,lambda = Lambda,tau = c(-1,2),Gamma = Gamma,dist = "SUN")
MC8 = MCmeanvarTMD(a,b,mu,Sigma,lambda = Lambda,tau = c(-1,2),Gamma = Gamma,nu = 1,dist = "SUT")

Cumulative distribution function for folded multivariate distributions

Description

It computes the cumulative distribution function on x for a folded p-variate Normal, Skew-normal (SN), Extended Skew-normal (ESN) and Student's t-distribution.

Usage

cdfFMD(x,mu,Sigma,lambda = NULL,tau = NULL,dist,nu = NULL)

Arguments

Details

Normal case by default, i.e., when dist is not provided. Univariate case is also considered, where Sigma will be the variance \sigma^2.

Value

It returns the distribution value for a single point x.

Note

Degrees of freedom must be a positive integer. If nu >= 200, Normal case is considered."

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

See Also

momentsFMD, meanvarFMD

Examples

mu = c(0.1,0.2,0.3,0.4)
Sigma = matrix(data = c(1,0.2,0.3,0.1,0.2,1,0.4,-0.1,0.3,0.4,1,0.2,0.1,-0.1,0.2,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
cdfFMD(x = c(0.5,0.2,1.0,1.3),mu,Sigma,dist="normal")
cdfFMD(x = c(0.5,0.2,1.0,1.3),mu,Sigma,dist = "t",nu = 4)
cdfFMD(x = c(0.5,0.2,1.0,1.3),mu,Sigma,lambda = c(-2,0,2,1),dist = "SN")
cdfFMD(x = c(0.5,0.2,1.0,1.3),mu,Sigma,lambda = c(-2,0,2,1),tau = 1,dist = "ESN")

Multivariate Extended-Skew Normal Density, Probablilities and Random Deviates Generator

Description

These functions provide the density function, probabilities and a random number generator for the multivariate extended-skew normal (ESN) distribution with mean vector mu, scale matrix Sigma, skewness parameter lambda and extension parameter tau.

Usage

dmvESN(x,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda,tau=0)
pmvESN(lower = rep(-Inf,length(lambda)),upper=rep(Inf,length(lambda)),
        mu = rep(0,length(lambda)),Sigma,lambda,tau,log2 = FALSE)
rmvESN(n,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda,tau=0)

Arguments

Value

dmvESN gives the density, pmvESN gives the distribution function, and rmvESN generates random deviates for the Multivariate Extended-Skew Normal Distribution.

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Galarza, C.E., Matos, L.A. and Lachos, V.H. (2022c). An EM algorithm for estimating the parameters of the multivariate skew-normal distribution with censored responses. Metron. <doi:10.1007/s40300-021-00227-4>.

Genz, A., (1992) "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149 <doi:10.1080/10618600.1992.10477010>.

See Also

dmvSN, pmvSN, rmvSN, meanvarFMD,meanvarTMD,momentsTMD

Examples

#Univariate case
dmvESN(x = -1,mu = 2,Sigma = 5,lambda = -2,tau = 0.5)
rmvESN(n = 100,mu = 2,Sigma = 5,lambda = -2,tau = 0.5)
#Multivariate case
mu = c(0.1,0.2,0.3,0.4)
Sigma = matrix(data = c(1,0.2,0.3,0.1,0.2,1,0.4,-0.1,0.3,0.4,1,0.2,0.1,-0.1,0.2,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
lambda = c(-2,0,1,2)
tau = 2
#One observation
dmvESN(x = c(-2,-1,0,1),mu,Sigma,lambda,tau)
rmvESN(n = 100,mu,Sigma,lambda,tau)
#Many observations as matrix
x = matrix(rnorm(4*10),ncol = 4,byrow = TRUE)
dmvESN(x = x,mu,Sigma,lambda,tau)

lower = rep(-Inf,4)
upper = c(-1,0,2,5)
pmvESN(lower,upper,mu,Sigma,lambda,tau)

Multivariate Extended-Skew t Density, Probablilities and Random Deviates Generator

Description

These functions provide the density function, probabilities and a random number generator for the multivariate extended-skew t (EST) distribution with mean vector mu, scale matrix Sigma, skewness parameter lambda, extension parameter tau and degrees of freedom nu.

Usage

dmvEST(x,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda,tau=0,nu)
pmvEST(lower = rep(-Inf,length(lambda)),upper=rep(Inf,length(lambda)),
        mu = rep(0,length(lambda)),Sigma,lambda,tau,nu,log2 = FALSE)
rmvEST(n,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda,tau,nu)

Arguments

Value

dmvEST gives the density, pmvEST gives the distribution function, and rmvEST generates random deviates for the Multivariate Extended-Skew-t Distribution.

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Genz, A., (1992) "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149 <doi:10.1080/10618600.1992.10477010>.

See Also

dmvST, pmvST, rmvST, meanvarFMD,meanvarTMD,momentsTMD

Examples

#Univariate case
dmvEST(x = -1,mu = 2,Sigma = 5,lambda = -2,tau = 0.5,nu=4)
rmvEST(n = 100,mu = 2,Sigma = 5,lambda = -2,tau = 0.5,nu=4)
#Multivariate case
mu = c(0.1,0.2,0.3,0.4)
Sigma = matrix(data = c(1,0.2,0.3,0.1,0.2,1,0.4,-0.1,0.3,0.4,1,0.2,0.1,-0.1,0.2,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
lambda = c(-2,0,1,2)
tau = 2
#One observation
dmvEST(x = c(-2,-1,0,1),mu,Sigma,lambda,tau,nu=4)
rmvEST(n = 100,mu,Sigma,lambda,tau,nu=4)
#Many observations as matrix
x = matrix(rnorm(4*10),ncol = 4,byrow = TRUE)
dmvEST(x = x,mu,Sigma,lambda,tau,nu=4)

lower = rep(-Inf,4)
upper = c(-1,0,2,5)
pmvEST(lower,upper,mu,Sigma,lambda,tau,nu=4)

Multivariate Skew Normal Density and Probabilities and Random Deviates

Description

These functions provide the density function and a random number generator for the multivariate skew normal (SN) distribution with mean vector mu, scale matrix Sigma and skewness parameter lambda.

Usage

dmvSN(x,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda)
pmvSN(lower = rep(-Inf,length(lambda)),upper=rep(Inf,length(lambda)),
        mu = rep(0,length(lambda)),Sigma,lambda,log2 = FALSE)
rmvSN(n,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda)

Arguments

Value

dmvSN gives the density, pmvSN gives the distribution function, and rmvSN generates random deviates for the Multivariate Skew-normal Distribution.

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Galarza, C.E., Matos, L.A. and Lachos, V.H. (2022c). An EM algorithm for estimating the parameters of the multivariate skew-normal distribution with censored responses. Metron. <doi:10.1007/s40300-021-00227-4>.

Genz, A., (1992) "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149 <doi:10.1080/10618600.1992.10477010>.

See Also

dmvESN, pmvESN, rmvESN, meanvarFMD,meanvarTMD,momentsTMD

Examples

#Univariate case
dmvSN(x = -1,mu = 2,Sigma = 5,lambda = -2)
rmvSN(n = 100,mu = 2,Sigma = 5,lambda = -2)
#Multivariate case
mu = c(0.1,0.2,0.3,0.4)
Sigma = matrix(data = c(1,0.2,0.3,0.1,0.2,1,0.4,-0.1,0.3,0.4,1,0.2,0.1,-0.1,0.2,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
lambda = c(-2,0,1,2)
#One observation
dmvSN(x = c(-2,-1,0,1),mu,Sigma,lambda)
rmvSN(n = 100,mu,Sigma,lambda)
#Many observations as matrix
x = matrix(rnorm(4*10),ncol = 4,byrow = TRUE)
dmvSN(x = x,mu,Sigma,lambda)

lower = rep(-Inf,4)
upper = c(-1,0,2,5)
pmvSN(lower,upper,mu,Sigma,lambda)

Multivariate Skew t Density, Probablilities and Random Deviates Generator

Description

These functions provide the density function, probabilities and a random number generator for the multivariate skew t (EST) distribution with mean vector mu, scale matrix Sigma, skewness parameter lambda and degrees of freedom nu.

Usage

dmvST(x,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda,nu)
pmvST(lower = rep(-Inf,length(lambda)),upper=rep(Inf,length(lambda)),
        mu = rep(0,length(lambda)),Sigma,lambda,nu,log2 = FALSE)
rmvST(n,mu=rep(0,length(lambda)),Sigma=diag(length(lambda)),lambda,nu)

Arguments

Value

dmvST gives the density, pmvST gives the distribution function, and rmvST generates random deviates for the Multivariate Skew-t Distribution.

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Genz, A., (1992) "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149 <doi:10.1080/10618600.1992.10477010>.

See Also

dmvST, pmvST, rmvST, meanvarFMD,meanvarTMD,momentsTMD

Examples

#Univariate case
dmvST(x = -1,mu = 2,Sigma = 5,lambda = -2,nu=4)
rmvST(n = 100,mu = 2,Sigma = 5,lambda = -2,nu=4)
#Multivariate case
mu = c(0.1,0.2,0.3,0.4)
Sigma = matrix(data = c(1,0.2,0.3,0.1,0.2,1,0.4,-0.1,0.3,0.4,1,0.2,0.1,-0.1,0.2,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
lambda = c(-2,0,1,2)
#One observation
dmvST(x = c(-2,-1,0,1),mu,Sigma,lambda,nu=4)
rmvST(n = 100,mu,Sigma,lambda,nu=4)
#Many observations as matrix
x = matrix(rnorm(4*10),ncol = 4,byrow = TRUE)
dmvST(x = x,mu,Sigma,lambda,nu=4)

lower = rep(-Inf,4)
upper = c(-1,0,2,5)
pmvST(lower,upper,mu,Sigma,lambda,nu=4)

Mean and variance for folded multivariate distributions

Description

It computes the mean vector and variance-covariance matrix for the folded p-variate Normal, Skew-normal (SN), Extended Skew-normal (ESN) and Student's t-distribution.

Usage

meanvarFMD(mu,Sigma,lambda = NULL,tau = NULL,nu = NULL,dist)

Arguments

Details

Normal case by default, i.e., when dist is not provided. Univariate case is also considered, where Sigma will be the variance \sigma^2.

Value

It returns a list with three elements:

Warning

The mean can only be provided when nu is larger than 2. On the other hand, the varcov matrix can only be provided when nu is larger than 3.

Note

Degree of freedom must be a positive integer. If nu >= 200, Normal case is considered."

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

See Also

momentsFMD, onlymeanTMD,meanvarTMD,momentsTMD, dmvSN,pmvSN,rmvSN, dmvESN,pmvESN,rmvESN, dmvST,pmvST,rmvST, dmvEST,pmvEST,rmvEST

Examples

mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
value1 = meanvarFMD(mu,Sigma,dist="normal")
value2 = meanvarFMD(mu,Sigma,nu = 4,dist = "t")
value3 = meanvarFMD(mu,Sigma,lambda = c(-2,0,1),dist = "SN")
value4 = meanvarFMD(mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")

Mean and variance for doubly truncated multivariate distributions

Description

It computes the mean vector and variance-covariance matrix for some doubly truncated skew-elliptical distributions. It supports the p-variate Normal, Skew-normal (SN), Extended Skew-normal (ESN) and Unified Skew-normal (SUN) as well as the Student's-t, Skew-t (ST), Extended Skew-t (EST) and Unified Skew-t (SUT) distribution.

Usage

meanvarTMD(lower = rep(-Inf,length(mu)),upper = rep(Inf,length(mu)),mu,Sigma
,lambda = NULL,tau = NULL,Gamma = NULL,nu = NULL,dist)

Arguments

Details

Univariate case is also considered, where Sigma will be the variance \sigma^2. Normal case code is an R adaptation of the Matlab available function dtmvnmom.m from Kan & Robotti (2017) and it is used for p<=3. For higher dimensions we use an extension of the algorithm in Vaida (2009).

Value

It returns a list with three elements:

Warning

For the t cases, the algorithm supports degrees of freedom nu <= 2.

Note

If nu >= 300, Normal case is considered."

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

See Also

MCmeanvarTMD, momentsTMD, meanvarFMD, meanvarFMD,momentsFMD, dmvSN,pmvSN,rmvSN, dmvESN,pmvESN,rmvESN, dmvST,pmvST,rmvST, dmvEST,pmvEST,rmvEST

Examples

a = c(-0.8,-0.7,-0.6)
b = c(0.5,0.6,0.7)
mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)

# Theoretical value
value1 = meanvarTMD(a,b,mu,Sigma,dist="normal")

#MC estimate
MC11 = MCmeanvarTMD(a,b,mu,Sigma,dist="normal") #by defalut n = 10000
MC12 = MCmeanvarTMD(a,b,mu,Sigma,dist="normal",n = 10^5) #more precision

# Now works for for any nu>0
value2 = meanvarTMD(a,b,mu,Sigma,dist = "t",nu = 0.87)

value3 = meanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),dist = "SN")
value4 = meanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),nu = 4,dist = "ST")
value5 = meanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")
value6 = meanvarTMD(a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,nu = 4,dist = "EST")


#Skew-unified Normal (SUN) and Skew-unified t (SUT) distributions

Lambda = matrix(c(1,0,2,-3,0,-1),3,2) #A skewness matrix p times q
Gamma  = matrix(c(1,-0.5,-0.5,1),2,2) #A correlation matrix q times q
tau    = c(-1,2)                      #A vector of extension parameters of dim q

value7 = meanvarTMD(a,b,mu,Sigma,lambda = Lambda,tau = c(-1,2),Gamma = Gamma,dist = "SUN")
value8 = meanvarTMD(a,b,mu,Sigma,lambda = Lambda,tau = c(-1,2),Gamma = Gamma,nu = 4,dist = "SUT")


#The ESN and EST as particular cases of the SUN and SUT for q=1

Lambda = matrix(c(-2,0,1),3,1)
Gamma  = 1
value9 = meanvarTMD(a,b,mu,Sigma,lambda = Lambda,tau = 1,Gamma = Gamma,dist = "SUN")
value10 = meanvarTMD(a,b,mu,Sigma,lambda = Lambda,tau = 1,Gamma = Gamma,nu = 4,dist = "SUT")

round(value5$varcov,2) == round(value9$varcov,2)
round(value6$varcov,2) == round(value10$varcov,2)

Moments for folded multivariate distributions

Description

It computes the kappa-th order moments for the folded p-variate Normal, Skew-normal (SN), Extended Skew-normal (ESN) and Student's t-distribution. It also output other lower moments involved in the recurrence approach.

Usage

momentsFMD(kappa,mu,Sigma,lambda = NULL,tau = NULL,nu = NULL,dist)

Arguments

Details

Univariate case is also considered, where Sigma will be the variance \sigma^2.

Value

A data frame containing p+1 columns. The p first containing the set of combinations of exponents summing up to kappa and the last column containing the the expected value. Normal cases (ESN, SN and normal) return prod(kappa)+1 moments while the Student's t-distribution case returns all moments of order up to kappa. See example section.

Warning

For the Student-t cases, including ST and EST, kappa-th order moments exist only for kappa < nu.

Note

Degrees of freedom must be a positive integer. If nu >= 300, Normal case is considered."

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850 <doi:10.1007/s00184-020-00802-1>.

Galarza, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

See Also

meanvarFMD, onlymeanTMD,meanvarTMD,momentsTMD, dmvSN,pmvSN,rmvSN, dmvESN,pmvESN,rmvESN, dmvST,pmvST,rmvST, dmvEST,pmvEST,rmvEST

Examples

mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
value1 = momentsFMD(c(2,0,1),mu,Sigma,dist="normal")
value2 = momentsFMD(3,mu,Sigma,dist = "t",nu = 7)
value3 = momentsFMD(c(2,0,1),mu,Sigma,lambda = c(-2,0,1),dist = "SN")
value4 = momentsFMD(c(2,0,1),mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")

#T case with kappa vector input
value5 = momentsFMD(c(2,0,1),mu,Sigma,dist = "t",nu = 7)

Moments for doubly truncated multivariate distributions

Description

It computes kappa-th order moments for for some doubly truncated skew-elliptical distributions. It supports the p-variate Normal, Skew-normal (SN) and Extended Skew-normal (ESN), as well as the Student's-t, Skew-t (ST) and the Extended Skew-t (EST) distribution.

Usage

momentsTMD(kappa,lower = rep(-Inf,length(mu)),upper = rep(Inf,length(mu)),mu,Sigma,
lambda = NULL,tau = NULL,nu = NULL,dist)

Arguments

Details

Univariate case is also considered, where Sigma will be the variance \sigma^2.

Value

A data frame containing p+1 columns. The p first containing the set of combinations of exponents summing up to kappa and the last column containing the the expected value. Normal cases (ESN, SN and normal) return prod(kappa)+1 moments while the Student's t-distribution case returns all moments of order up to kappa. See example section.

Note

If nu >= 300, Normal case is considered."

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850.

Galarza-Morales, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Kan, R., & Robotti, C. (2017). On moments of folded and truncated multivariate normal distributions. Journal of Computational and Graphical Statistics, 26(4), 930-934.

See Also

onlymeanTMD,meanvarTMD,momentsFMD,meanvarFMD,dmvSN,pmvSN,rmvSN, dmvESN,pmvESN,rmvESN, dmvST,pmvST,rmvST, dmvEST,pmvEST,rmvEST

Examples

a = c(-0.8,-0.7,-0.6)
b = c(0.5,0.6,0.7)
mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
value1 = momentsTMD(c(2,0,1),a,b,mu,Sigma,dist="normal")
value2 = momentsTMD(c(2,0,1),a,b,mu,Sigma,dist = "t",nu = 7)
value3 = momentsTMD(c(2,0,1),a,b,mu,Sigma,lambda = c(-2,0,1),dist = "SN")
value4 = momentsTMD(c(2,0,1),a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")


#T cases with kappa scalar (all moments up to 3)
value5 = momentsTMD(3,a,b,mu,Sigma,nu = 7,dist = "t")
value6 = momentsTMD(3,a,b,mu,Sigma,lambda = c(-2,0,1),nu = 7,dist = "ST")
value7 = momentsTMD(3,a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,nu = 7,dist = "EST")

Mean for doubly truncated multivariate distributions

Description

It computes the mean vector for some doubly truncated skew-elliptical distributions. It supports the p-variate Normal, Skew-normal (SN), Extended Skew-normal (ESN) and Unified Skew-normal (SUN) as well as the Student's-t, Skew-t (ST), Extended Skew-t (EST) and Unified Skew-t (SUT) distribution.

Usage

onlymeanTMD(lower = rep(-Inf, length(mu)),upper = rep(Inf,length(mu)),mu,Sigma,
             lambda = NULL,tau = NULL,Gamma = NULL,nu = NULL,dist)

Arguments

Details

Univariate case is also considered, where Sigma will be the variance \sigma^2. Normal case code is an R adaptation of the Matlab available function dtmvnmom.m from Kan & Robotti (2017) and it is used for p<=3. For higher dimensions we use proposal in Galarza (2022b).

Value

It returns the mean vector of length p.

Note

Degrees of freedom must be a positive integer. If nu >= 300, Normal case is considered."

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Galarza, C. E., Lin, T. I., Wang, W. L., & Lachos, V. H. (2021). On moments of folded and truncated multivariate Student-t distributions based on recurrence relations. Metrika, 84(6), 825-850.

Galarza-Morales, C. E., Matos, L. A., Dey, D. K., & Lachos, V. H. (2022a). "On moments of folded and doubly truncated multivariate extended skew-normal distributions." Journal of Computational and Graphical Statistics, 1-11 <doi:10.1080/10618600.2021.2000869>.

Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022b). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>.

Kan, R., & Robotti, C. (2017). On moments of folded and truncated multivariate normal distributions. Journal of Computational and Graphical Statistics, 26(4), 930-934.

See Also

momentsTMD, meanvarFMD, momentsFMD,dmvESN,rmvESN

Examples

a = c(-0.8,-0.7,-0.6)
b = c(0.5,0.6,0.7)
mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)
value1 = onlymeanTMD(a,b,mu,Sigma,dist="normal")

# Now works for for any nu>0
value2 = onlymeanTMD(a,b,mu,Sigma,dist = "t",nu = 0.87)

value3 = onlymeanTMD(a,b,mu,Sigma,lambda = c(-2,0,1),dist = "SN")
value4 = onlymeanTMD(a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")
value5 = onlymeanTMD(a,b,mu,Sigma,lambda = c(-2,0,1),tau = 1,nu = 4,dist = "EST")

#Skew-unified Normal (SUN) and Skew-unified t (SUT) distributions

Lambda = matrix(c(1,0,2,-3,0,-1),3,2) #A skewness matrix p times q
Gamma  = matrix(c(1,-0.5,-0.5,1),2,2) #A correlation matrix q times q
tau    = c(-1,2)                      #A vector of extension parameters of dim q

value6 = onlymeanTMD(a,b,mu,Sigma,lambda = Lambda,tau = c(-1,2),Gamma = Gamma,dist = "SUN")
value7 = onlymeanTMD(a,b,mu,Sigma,lambda = Lambda,tau = c(-1,2),Gamma = Gamma,nu = 4,dist = "SUT")

#The ESN and EST as particular cases of the SUN and SUT for q=1

Lambda = matrix(c(-2,0,1),3,1)
Gamma  = 1
value8 = onlymeanTMD(a,b,mu,Sigma,lambda = Lambda,tau = 1,Gamma = Gamma,dist = "SUN")
value9 = onlymeanTMD(a,b,mu,Sigma,lambda = Lambda,tau = 1,Gamma = Gamma,nu = 4,dist = "SUT")

round(value4,2) == round(value8,2)
round(value5,2) == round(value9,2)

Multivariate normal and Student-t probabilities

Description

Computation of Multivariate normal and Student-t probabilities using the classic Genz method form packages mvtnorm and tlrmvnmvt packages. In order to save computational effort, it chooses whether to use the function pmvtnorm (pmvt) from mvtnorm, or functions pmvn (pmvt) from the tlrmvnmvt package, depending of the vector size p, real or integer degrees of freedom nu.

Usage

pmvnormt(lower = rep(-Inf,ncol(sigma)),upper = rep(Inf,ncol(sigma)),
mean = rep(0,ncol(sigma)),sigma,nu = NULL,uselog2 = FALSE)

Arguments

Value

The estimated probability or its log2 if uselog2 == TRUE

Note

If is.null(nu), normal case is considered.

Author(s)

Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@uconn.edu>

Maintainer: Christian E. Galarza <cgalarza88@gmail.com>

References

Genz, A. (1992), "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics, 1, 141-149.

Cao, J., Genton, M. G., Keyes, D. E., & Turkiyyah, G. M. "Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student- t Probabilities" (2019) <https://marcgenton.github.io/2019.CGKT.manuscript.pdf>

See Also

onlymeanTMD,meanvarTMD,momentsFMD,momentsTMD,meanvarFMD,dmvSN,pmvSN,rmvSN, dmvESN,pmvESN,rmvESN, dmvST,pmvST,rmvST, dmvEST,pmvEST,rmvEST

Examples

a = c(-0.8,-0.7,-0.6)
b = c(0.5,0.6,0.7)
mu = c(0.1,0.2,0.3)
Sigma = matrix(data = c(1,0.2,0.3,0.2,1,0.4,0.3,0.4,1),
               nrow = length(mu),ncol = length(mu),byrow = TRUE)

pmvnormt(lower = a,upper = b,mean = mu,sigma = Sigma) #normal case
pmvnormt(lower = a,upper = b,mean = mu,sigma = Sigma,nu = 4.23)  #t case 
pmvnormt(lower = a,upper = b,mean = mu,sigma = Sigma,nu = 4.23,uselog2 = TRUE)