通过R中的自定义分布进行仿真

Question

Does anyone know how to simulate random variables from the following probability density function (pdf):

where g(x) is the pdf for standard normal distribution N(0,1). I understand that simulating random variables from customised distribution involves finding the cumulative density function and using the uniform distribution (as there were similar questions on this topic in this platform also). However, the pdf I have here looks a bit more complicated than the other examples I came across before. For example, I would imagine that finding the cumulative density function (need to integrate) is not so straight forward. Can anyone give me advice on how to solve this? Or is there another simpler method to simulate from the above distribution? Look forward to your advice. Thanks!

Answer 1

You might try a Markov chain Monte Carlo approach such as the Metropolis-Hastings Algorithm. https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm

Answer 2

Doesn't this work?

special <- function(x) {
               if (x <= -1) {
                  val <- dnorm(x) / (1+exp(-x-1))
               } else if ((x > -1) & (x < 1)) {
                  val <- dnorm(x) / 2
               } else {
                  val <- dnorm(x) / (1+exp(-x+1))
               }
               return(val)
           }

special(-2)
[1] 0.01452041

special(-0.5)
[1] 0.1760327

Answer 3

You can do this using sample ...

#define some x values
x <- seq(-5, 5, 0.001)
#define the pdf as above
f <- function(x) {
  if (x <= -1) {
    y <- dnorm(x)/(1+exp(-x-1))
  } else if (x >= 1) {
    y <- dnorm(x)/(1+exp(-x+1))
  } else {
    y <- dnorm(x)/2
  }
  return(y)
}
#calculate values of pdf for all x
fx <- f(x)
#create sample using these probabilities
samp <- sample(x, size=10000, replace=TRUE, prob=fx)

hist(samp, breaks=50)