momentsTMD function

Moments for doubly truncated multivariate distributions

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.

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

Arguments

• kappa: moments vector of length $p$. All its elements must be integers greater or equal to $0$. For the Student's-t case, kappa can be a scalar representing the order of the moment.
• lower: the vector of lower limits of length $p$.
• upper: the vector of upper limits of length $p$.
• mu: a numeric vector of length $p$ representing the location parameter.
• Sigma: a numeric positive definite matrix with dimension $p$x$p$ representing the scale parameter.
• lambda: a numeric vector of length $p$ representing the skewness parameter for SN and ESN cases. If lambda == 0, the ESN/SN reduces to a normal (symmetric) distribution.
• tau: It represents the extension parameter for the ESN distribution. If tau == 0, the ESN reduces to a SN distribution.
• nu: It represents the degrees of freedom for the Student's t-distribution being a positive real number.
• dist: represents the truncated distribution to be used. The values are normal, SN and ESN for the doubly truncated Normal, Skew-normal and Extended Skew-normal distributions and, t, ST and EST for the for the doubly truncated Student-t, Skew-t and Extended Skew-t distributions.

Details

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

Returns

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.

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.

Author(s)

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

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

Note

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

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")