Integration in R with integrate function

Question

library(pbivnorm)

rho <- 0.5
f1 <- function(x, y) {
  pbivnorm(log(x)-10, log(y)-10, rho)*(exp(-(log(x)-10)^2/2)/(sqrt(2*pi)*x))*(exp(-(log(y)-10)^2/2)/(sqrt(2*pi)*y))
}
integration1 <- round(integrate(function(y) {
  sapply(y, function(y) {
    integrate(function(x) f1(x,y), 0, Inf, rel.tol = 1e-12)$value
  })
}, 0, Inf, rel.tol = 1e-12)$value, 10)

This integration should be around 0.3, but R gives 0. Could anyone point out the problem? What is the best function for integral in R? Many thanks.

Answer 1

Package cubature can solve the problem giving the expected result. The function must be rewritten as a one argument function, and the values for x and y set in the function body.

library(cubature)

f2 <- function(X) {
  x <- X[1]
  y <- X[2]
  pbivnorm(log(x)-10, log(y)-10, rho)*(exp(-(log(x)-10)^2/2)/(sqrt(2*pi)*x))*(exp(-(log(y)-10)^2/2)/(sqrt(2*pi)*y))
}

hcubature(f2, c(0, 0), c(Inf, Inf))
#$integral
#[1] 0.2902153
#
#$error
#[1] 2.863613e-06
#
#$functionEvaluations
#[1] 7599
#
#$returnCode
#[1] 0

Edit.

Following the OP's second comment, here is the integral computed with hcubature

f3 <- function(x) {
  pnorm(log(x)-10.2)*(exp(-(log(x)-10)^2/2)/(sqrt(2*pi)*x)) 
} 

hcubature(f3, lowerLimit = 0, upperLimit = Inf, tol = 1e-12)$integral 
#[1] 0.4437685