Acceptance-Rejection method to generate random numbers from Gamma

Question

How to use the Acceptance-Rejection (A-R) method to generate random numbers from the following gamma distribution with density

$$f(x) = \frac1{4^3Γ(3)}x^2\exp^{-\frac x4}, \ x > 0,$$

where $\Gamma(·)$ is the gamma function. The proposal distribution of your choice has to be a distribution from which you can easily generate random numbers using probability integral transformation method. For the proposal distribution of your choice, find the best possible constant C and the corresponding acceptance probability of the A-R method.

I'm not sure how to go about finding a proposal distribution or finding an C (or M depending on what text you read)

Answer 1

Here is a start. In order to use the A-R method to simulate a distribution with density $f(x),$ you need a function that is an appropriate constant multiple $C$ of the PDF $g(x)$ of an easily simulated distribution, such that $Cg(x) \ge f(x)$ throughout the support of $f(x).$

Here is a sketch of the density of $\mathsf{Gamma}(shape=3, rate=1/4)$ along with a sketch of double the density of $\mathsf{Exp}(rate=1/12).$

R code for the sketch:

curve(dgamma(x, 3, 1/4), 0, 30, ylim=c(0,.2), lwd=2, col="blue")
curve(2*dexp(x,1/12), 0, 30, add=T, col="red", lwd=2)
abline(v=0, col="green2");  abline(h=0, col="green2")

The exponential distribution is easily simulated from the standard uniform distribution via the quantile method. Below about $x=8,$ the red curve is much higher than the blue one, but from there on the fit isn't bad. Obviously, the red curve is a little higher than it needs to be, so $C=2$ is not quite the optimal constant.

I am not saying this is the best possible choice, but it illustrates the kind of thinking you need to do, and I hope the sketch is helpful.