Question

I would like to solve an equation as below, where the X is the only unknown variable and function f() is a multi-variate Student t distribution.
More precisely, I have a multi k-dimensional integral for a student density function, which gives us a probability as a result, and I know that this probability is given as q. The lower bound for all integral is -Inf and I know the last k-1 dimension’s upper bound (as given), the only unknown variable is the first integral’s upper bound. It should have an solution for a variable and one equation. I tried to solve it in R. I did Dynamic Conditional Correlation to have a correlation matrix in order to specify my t-distribution. So plug this correlation matrix into my multi t distribution “dmvt”, and use the “adaptIntegral” function from “cubature” package to construct a function as an argument to the command “uniroot” to solve the upper bound on the first integral. But I have some difficulties to achieve what I want to get. (I hope my question is clear) I have provided my codes before, somebody told me that there is problem, but cannot find why there is an issue there. Many thanks in advance for your help.

I now how to deal with it with one dimension integral, but I don’t know how a multi-dimension integral equation can be solved in R? (e.g. for 2 dimension case)

\int_{-\infty}^{X}
    \int_{-\infty}^{Y_{1}} \cdots
       \int_{-\infty}^{Y_{k}}
           f(x,y_{1},\cdots y_{k})
      d_{x}d_{y_{1},}\cdots d_{y_{k}} = q

This code fails:

require(cubature) 
require(mvtnorm) 
corr <- matrix(c(1,0.8,0.8,1),2,2)
f <- function(x){ dmvt(x,sigma=corr,df=3) } 
g <- function(y) adaptIntegrate(f, 
                  lowerLimit = c( -Inf, -Inf), 
                  upperLimit = c(y, -0.1023071))$integral-0.0001    
uniroot( g, c(-2, 2)) 

Answer 1

Since mvtnorm includes a pmvt function that computes the CDF of the multivariate t distribution, you don’t need to do the integral by brute force. (mvtnorm also includes a quantile function qmvt, but only for “equicoordinate” values.)

So:

library(mvtnorm)
g <- function(y1_upr,y2_upr=-0.123071,target=1e-4,df=3) {
    pmvt(upper=c(y1_upr,y2_upr),df=df)-target
}
uniroot(g,c(-10000,0))
## $root
## [1] -17.55139
## 
## $f.root
## [1] -1.699876e-11
## attr(,"error")
## [1] 1e-15
## attr(,"msg")
## [1] "Normal Completion"
## 
## $iter
## [1] 18
## 
## $estim.prec
## [1] 6.103516e-05
## 

Double-check:

pmvt(upper=c(-17.55139,-0.123071),df=3)
## [1] 1e-04
## attr(,"error")
## [1] 1e-15
## attr(,"msg")
## [1] "Normal Completion"