meanvarFMD function

Mean and variance for folded multivariate distributions

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.

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

Arguments

• 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. Must be an integer greater than 1.
• dist: represents the folded distribution to be computed. The values are normal, SN , ESN and t for the doubly truncated Normal, Skew-normal, Extended Skew-normal and Student's t-distribution respectively.

Details

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

Returns

It returns a list with three elements: - mean: the mean vector of length $p$

• EYY: the second moment matrix of dimensions $p$x$p$

• varcov: the variance-covariance matrix of dimensions $p$x$p$

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.

Author(s)

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

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

Note

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

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.

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")