The integral is probably divergent - r

Question

fun1 = function(y,mu=mu0,lsig=lsig0) {
res = 1/(exp(-y)+1)^2 * 1/sqrt(2*pi)/exp(lsig) * exp(-(y-mu)^2/2/exp(lsig)^2)
return(res)
}
fun4 = function(para=c(mu1,lsig1)) {
mu1 = para[1]
lsig1 = para[2]
res = n1 * log(noze(integrate(fun1,-Inf,Inf,mu=mu1,lsig=lsig1)$value)) +
n3 * log(noze(integrate(fun2,-Inf,Inf,mu=mu1,lsig=lsig1)$value)) +
n2 * log(noze(integrate(fun3,-Inf,Inf,mu=mu1,lsig=lsig1)$value))
return(-res)
}
noze = function(x) {
if (x < 1e-100) { x = 1e-100 }
return(x)
}
optim(c(0.5,2),fun4,method="L-BFGS-B",lower=c(-5,-3),upper=c(3.5,3.5))$par
I have to find two parameters of function 'fun4' which uses the integral of 'fun1.' ('fun2' and 'fun3' are slightly different from 'fun1')
I encountered an error 'Error in integrate(fun1, -Inf, Inf, -3.9538, -3) :
the integral is probably divergent'
Using scatterplot, I found that fun1 is close to zero almost everywhere except for (-4.2,-3.7).
Thus, integrating for that interval only gives (approximately) correct integral.
> integrate(fun1,-4.2,-3.6,-3.9538,-3)
0.0003558953 with absolute error < 3e-11
This can be confirmed using nearby parameter values
> integrate(fun1,-Inf,Inf,-3.9538,-3.1)
0.0003555906 with absolute error < 2.6e-05
> integrate(fun1,-Inf,Inf,-3.9538,-2.85)
0.0003564842 with absolute error < 3.7e-06
If the interval is too wide, it gives incorrect integral.
> integrate(fun1,-5,5,-3.9538,-3)
0.0003558953 with absolute error < 2.3e-08
> integrate(fun1,-15,15,-3.9538,-3)
3.492547e-11 with absolute error < 6.5e-11
> integrate(fun1,-30,30,-3.9538,-3)
1.980146e-41 with absolute error < 3.4e-41
> integrate(fun1,-50,50,-3.9538,-3)
0 with absolute error < 0
> integrate(fun1,-Inf,Inf,-3.9538,-3)
Error in integrate(fun1, -Inf, Inf, -3.9538, -3) :
the integral is probably divergent
If I have to integrate only once, I can find an interval where 'fun1' is large enough and integrate only for that interval.
But the problem is I use optim function which tries various parameters to find a minimizer of 'fun4.'
Using (-Inf,Inf) gives an error and wide enough interval gives incorrect integrals.
Is there a good method to solve this?

Answer 1

Convolution with gaussian kernel might be solved using Gauss-Hermite integration, and there is R package for that: https://cran.r-project.org/web/packages/gaussquad/gaussquad.pdf
Some test code:
library(gaussquad)
n.quad <- 128 # integration order
# get the particular weights/abscissas as data frame with 2 observables and n.quad observations
rule <- ghermite.h.quadrature.rules(n.quad, mu = 0.0)[[n.quad]]
# test function - integrate 1 over exp(-x^2) from -Inf to Inf
# should get sqrt(pi) as an answer
f <- function(x) {
1.0
}
q <- ghermite.h.quadrature(f, rule)
print(q - sqrt(pi))