MCmeanvarTMD function

Monte Carlo Mean and variance for doubly truncated multivariate distributions

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.

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

• 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 matrix of dimension $p$x$q$ representing the skewness/shape matrix parameter for the SUN and SUT distribution. For the ESN and EST distributions ($q=1$), lambda is a numeric vector of dimension $p$ (see examples at the end of this help). If all(lambda == 0), the SUN/ESN/SN (SUT/EST/ST) reduces to a normal (t) symmetric distribution.
• tau: a numeric vector of length $q$ representing the extension parameter for the SUN and SUT distribution. For the ESN and EST distributions, tau is a positive scalar ($q=1$). Furthermore, if tau == 0, the ESN (EST) reduces to a SN (ST) distribution.
• Gamma: a correlation matrix with dimension $q$x$q$. It must be provided only for the SUN and SUT cases. For particular cases SN, ESN, ST and EST, we have that Gamma == 1 (see examples at the end of this help).
• 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 , ESN and SUN for the doubly truncated Normal, Skew-normal, Extended Skew-normal and Unified-skew normal distributions and, t, ST , EST and SUT for the for the doubly truncated Student-t, Skew-t, Extended Skew-t and Unified skew-t distributions.
• n: number of Monte Carlo samples to be generated.

Returns

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

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

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

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.

Author(s)

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

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

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")