Moments of Folded and Doubly Truncated Multivariate Distributions.
Computation of truncated moments using the MomTrunc R package
The MomTrunc R package computes arbitrary products moments (mean vector and variance-covariance matrix), for some doubly 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. Methods for computing these moments are based on this seminal work.
Next, we will show some useful functions related to some members of this family, which includes the extended skew-t (EST) and its particular cases, those are, the extended skew-normal (ESN), the skew-t (ST), the skew-normal (SN), symmetric Student’s-t (MVT) and symmetric normal (MVN).
Density, probabilities and random deviates
These can be reached in the same fashion that other R base distributions, that is, using d, p and r followed by the distribution string name to get the PDF, CDF and random generating functions, respectively.
For instance, the extended skew-normal distribution, density values can be computed using dmvESN() probabilities with pmvESN(), and rmvESN() functions return generation of random variables from our distributions of interest.
Available string names are shown in the table below.
| Distribution | Option | String |
|---|---|---|
| Skew-normal | d, p, r | mvSN |
| Skew-t | d, p, r | mvST |
| Extended Skew-normal | d, p, r | mvESN |
| Extended Skew-t | d, p, r | mvEST |
library(MomTrunc)
#Univariate EST case
dmvESN(x = -1,mu = 2,Sigma = 5,lambda = -2,tau = 0.5)
#> [1] 0.1231744
sample = rmvEST(n = 1e5,mu = 2,Sigma = 5,lambda = -2,tau = 0.5,nu = 4)
print(head(sample))
#> [1] -0.5015923 -0.6642021 0.2891897 3.5957041 2.9828134 3.4525520
#plotting
hist(sample,breaks = 150,freq = FALSE,xlim = c(-15,10),main = "Histogram of EST variates")
curve(expr = dmvEST(x,mu = 2,Sigma = 5,lambda = -2,tau = 0.5,nu = 4),
from = -15,to = 10,n = 200,lwd = 2,col = 4,add = TRUE)
#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) #Skew-normal
#> [1] 0.003279037
rmvST(n = 10,mu,Sigma,lambda,nu = 2) #Skew-t
#> [,1] [,2] [,3] [,4]
#> [1,] 0.35393287 0.18670145 0.7951902 0.3964539
#> [2,] -1.22145843 1.98821902 2.2555133 2.4771210
#> [3,] 0.11505972 0.18907948 0.2638879 0.4822362
#> [4,] -0.79663450 1.12417008 1.2132558 -0.3534125
#> [5,] -0.27912920 0.17194797 0.1671234 0.3730229
#> [6,] 0.27243028 -0.01265475 0.9086517 0.3656542
#> [7,] 0.41908397 0.77848822 0.3260464 0.7753323
#> [8,] 0.78939078 0.74624772 1.3954601 0.5453789
#> [9,] 0.03073276 0.65275698 1.2502771 0.3931312
#> [10,] 0.47251536 0.17043266 1.7512018 0.7576431
#Many observations as matrix
x = matrix(rnorm(4*10),ncol = 4,byrow = TRUE)
dmvST(x = x,mu,Sigma,lambda,nu = 2) #Skew-t
#> [1] 7.255175e-07 2.994456e-04 3.493918e-03 3.356577e-06 2.428353e-03
#> [6] 3.762044e-05 2.284900e-02 1.217553e-02 3.003915e-02 1.311547e-06
# Probability between some points
lower = rep(-5,4)
upper = c(-1,0,2,5)
pmvSN(lower,upper,mu,Sigma,lambda) #Skew-normal
#> [1] 0.123428
pmvST(lower,upper,mu,Sigma,lambda,nu=2) #Skew-t
#> [1] 0.1335012
The pmvSN() and pmvESN() functions 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 directly to our C++ functions) and Cao et al. (2019) through the package tlrmvnmvt.
Mean and variance-covariance matrix
For this purpose, we call the function meanvarTMD() which returns the mean vector and variance-covariance matrix for some doubly truncated skew-elliptical distributions. It supports the -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. The distribution to be used is set by the argument dist. Next, we present some sample codes.
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)
#> [,1] [,2] [,3]
#> [1,] TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE
#> [3,] TRUE TRUE TRUE
round(value6$varcov,2) == round(value10$varcov,2)
#> [,1] [,2] [,3]
#> [1,] TRUE TRUE TRUE
#> [2,] TRUE TRUE TRUE
#> [3,] TRUE TRUE TRUE
As noted in the codes above, it is possible to obtain the moments by Monte Carlo approximation through the MCmeanvarTMD() function.
High-order moments
Finally, the momentsTMD provides the product moment for some truncated multivariate distributions. For instance, in order to compute the moment 𝔼[Y13Y21Y32 | a1≤Y1≤b1, a2≤Y2≤b2, a3≤Y3≤b3], for Y = (Y1,Y2,Y3)⊤ ∼ ESN3(μ,Σ,λ,τ), we run the following code:
momentsTMD(kappa = c(3,1,2),lower = a,upper = b,mu,Sigma,lambda = c(-2,0,1),tau = 1,dist = "ESN")
#> k1 k2 k3 E[k]
#> 1 0 0 0 1.0000000000
#> 2 1 0 0 -0.1955214032
#> 3 2 0 0 0.1604269300
#> 4 3 0 0 -0.0737276819
#> 5 0 1 0 -0.0284407326
#> 6 1 1 0 0.0075618650
#> 7 2 1 0 -0.0052312893
#> 8 3 1 0 0.0027692663
#> 9 0 0 1 0.1125134640
#> 10 1 0 1 -0.0041546757
#> 11 2 0 1 0.0130889137
#> 12 3 0 1 -0.0030069873
#> 13 0 1 1 0.0048928388
#> 14 1 1 1 -0.0012302466
#> 15 2 1 1 0.0008848539
#> 16 3 1 1 -0.0004097346
#> 17 0 0 2 0.1390876665
#> 18 1 0 2 -0.0249750438
#> 19 2 0 2 0.0219172190
#> 20 3 0 2 -0.0096157104
#> 21 0 1 2 -0.0026900254
#> 22 1 1 2 0.0008924818
#> 23 2 1 2 -0.0005106163
#> 24 3 1 2 0.0003672320
Note that some other lower order moments involved in the computation are also returned.
Folded distribution
Functions for the folded cases are also offered to the users. The analogous functions meanvarFMD(), momentsFMD() are used for the mean and variance-covariance matrix, and arbitrary product moments, respectively. Besides, the cdfFMD() computes the cdf. The available distributions are normal, Student-t, SN and ESN being set by the argument dist. Some sample codes are shown next.
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)
#cdf
cdfFMD(x = c(0.5,0.2,1.0,1.3),mu,Sigma,lambda = c(-2,0,2,1),dist = "SN")
#> [1] 0.02794654
#Mean and variance-covariance matrix
meanvarFMD(mu,Sigma,dist = "t",nu = 4)
#> $mean
#> [,1]
#> [1,] 1.003746
#> [2,] 1.014938
#> [3,] 1.033438
#> [4,] 1.059027
#>
#> $EYY
#> [,1] [,2] [,3] [,4]
#> [1,] 2.010000 1.316949 1.367027 1.335528
#> [2,] 1.316949 2.040000 1.430244 1.338320
#> [3,] 1.367027 1.430244 2.090000 1.392964
#> [4,] 1.335528 1.338320 1.392964 2.160000
#>
#> $varcov
#> [,1] [,2] [,3] [,4]
#> [1,] 1.0024938 0.2982090 0.3297167 0.2725335
#> [2,] 0.2982090 1.0099010 0.3813678 0.2634737
#> [3,] 0.3297167 0.3813678 1.0220049 0.2985250
#> [4,] 0.2725335 0.2634737 0.2985250 1.0384615
#Product moment c(2,0,1,2)
momentsFMD(kappa = c(2,0,1,2),mu,Sigma,lambda = c(-2,0,2,1),tau = 1,dist = "ESN")
#> k1 k2 k3 k4 E[k]
#> 1 2 0 1 2 1.3147576
#> 2 1 0 1 2 1.0309879
#> 3 0 0 1 2 1.2496227
#> 4 2 0 0 2 1.1854733
#> 5 1 0 0 2 0.9932095
#> 6 0 0 0 2 1.3074975
#> 7 2 0 1 1 0.8707904
#> 8 1 0 1 1 0.6921804
#> 9 0 0 1 1 0.8518643
#> 10 2 0 0 1 0.8261674
#> 11 1 0 0 1 0.6949156
#> 12 0 0 0 1 0.9196535
#> 13 2 0 1 0 0.8847480
#> 14 1 0 1 0 0.7128806
#> 15 0 0 1 0 0.8925955
#> 16 2 0 0 0 0.8956343
#> 17 1 0 0 0 0.7535822
#> 18 0 0 0 0 1.0000000
References
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
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.
Genz, A. (1992), “Numerical computation of multivariate normal probabilities,” Journal of Computational and Graphical Statistics, 1, 141-149.
Kan, R., & Robotti, C. (2017). On moments of folded and truncated multivariate normal distributions. Journal of Computational and Graphical Statistics, 26(4), 930-934.