The Pareto distribution is a heavy tailed distribution, suitable as candidate for modeling large insurance losses above a threshold. It is a mixture of exponential distributions with Gamma mixing weights. Another way to generate the Pareto distribution is taking the inverse of another distribution (raising another distribution to the power of minus one). Previous discussion on the Pareto distribution can be found here: An example of a mixture and The Pareto distribution.
Let be a continuous random variable with pdf and with cdf such that . Let . Then the resulting distribution for is called an inverse. For example, if has an exponential distribution, then is said to have an inverse exponential distribution. The following are the cdf and pdf of .
We now show that the Pareto distribution is an inverse. Suppose that has the pdf . We show that has a Pareto distribution with scale paramter and shape parameter . Once this base distribution is established, we can relax the scale parameter to have other positive values.
The cdf of is where . Since the support of is , the support of is . Thus in deriving the cdf , we only need to focus on (or ). The following is the cdf :
Upon differentiation, we obtain the pdf:
The above pdf is that of a Pareto distribution with scale paramter and shape parameter . However, the support of this pdf is . In order to have as the support, we have the following alternative version:
We now transform the above pdf to become a true 2-parameter Pareto pdf by relaxing the scale parameter. The result is the following pdf.