如何从 R 中已知密度函数的分布中生成随机值？

Question

I am given Zi's (i = 1,2...) form a sequence of iid variables with density function:

fZ(z) = P(S1 > z)/E(S1)

Given that i know S1 follows a Pareto(1.7,70) distribution and E(S1) is equal to 100, how would i use R to obtain/generate random values of Z? Like we all know how to generate random values of a normal distribution by using rnorm(), but what if our density was custom such as above? How would I simulate values that fit that distribution?

Thank you

Answer 1

Let us start with the Pareto(1.7,70) .

library(EnvStats)

custom_density <- function(z, location, shape) {
  stopifnot(shape > 1)
  ExpectS <- location * shape
  ExpectS <- ExpectS / (shape - 1)
  #' this should be equal to 100
  # print(ExpectS)

  (1 - EnvStats::ppareto(z, location = location, shape = shape)) /
    ExpectS
    # 100
}

custom_density(10, shape = 1.7, location = 70)

This is how I'm interpreting your custom density.

plot.new()
curve(
  custom_density(x, shape = 1.7, location = 70),
  xlim = c(-2500,1000))
curve(dgamma(x, shape = 0.6, scale = 300), xlim = c(0, 1000), add = TRUE)

This is how the custom density looks like, and also we can relate it to something more common, like the gamma. I picked parameters until the two curves were similar enough.

Then I decided on rejection-sampling as the way to sample from this.

There is a package that facilitates this {AR} :

library(AR)

AR::AR.Sim(
  100,
  f_X = function(x) custom_density(x, shape = 1.7, location = 70),
  Y.dist = "gamma",
  Y.dist.par = c(shape = 0.6, rate = 1/300)
)