Monte Carlo simulation to generate random numbers from Pareto distribution

Question

I am trying to generate random numbers using Monte Carlo simulation from Pareto distribution using R. But I am not able to find the codes for the same. It would be very helpful if someone could share some link for the R codes or some reading material.

Answer 1

According to Wikipedia, the cumulative density function (CDF) of a Pareto distribution has the form

$$F(x;x_m,\alpha) = 1 - \left(\frac{x_m}{x}\right)^\alpha$$

for positive numbers $x_m$ and $\alpha.$

This is simple to invert because the rules of algebra permit us to solve the equation $u=F(x;x_m,\alpha)$ as

$$x = \frac{x_m}{(1-u)^{1/\alpha}}.$$

Because $F$ is differentiable in $x,$ it is a continuous distribution, whence by letting $U$ have a uniform distribution on the interval $[0,1)$ and substituting it for $u,$ we see that the random variable

$$X = \frac{x_m}{(1-U)^{1/\alpha}}$$

has the desired Pareto distribution. (This is an application of the Probability Integral Transform.) Note that since $U$ and $1-U$ have the same distribution, the denominator may be simplified to $U.$

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The R function runif draws uniform random variates on the interval $[0,1).$ Thus, a direct R implementation is

rpareto <- function(n, alpha, x.m) x.m / runif(n)^(1/alpha)

So little computation is needed that we may conclude this is an efficient implementation.