Another way to generate the Pareto distribution

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. $\text{ }$

Let $X$ be a continuous random variable with pdf $f_X(x)$ and with cdf $F_X(x)$ such that $F_X(0)=0$. Let $Y=X^{-1}$. Then the resulting distribution for $Y$ is called an inverse. For example, if $X$ has an exponential distribution, then $Y$ is said to have an inverse exponential distribution. The following are the cdf and pdf of $Y=X^{-1}$. $\text{ }$ $\displaystyle F_Y(y)=1-F_X(y^{-1}) \ \ \ \ \ \ \ \ f_Y(y)=f_X(y^{-1}) \ y^{-2}$ $\text{ }$

We now show that the Pareto distribution is an inverse. Suppose that $X$ has the pdf $f_X(x)=\beta \ x^{\beta-1}, 0. We show that $Y=X^{-1}$ has a Pareto distribution with scale paramter $\alpha=1$ and shape parameter $\beta>0$. Once this base distribution is established, we can relax the scale parameter to have other positive values. $\text{ }$

The cdf of $X$ is $F_X(x)=x^\beta$ where $0. Since the support of $X$ is $0, the support of $Y$ is $y>1$. Thus in deriving the cdf $F_Y(y)$, we only need to focus on $y>1$ (or $0). The following is the cdf $F_Y(y)$: $\text{ }$ $\displaystyle F_Y(y)=1-F_X(y^{-1})=1-y^{-\beta}, \ \ \ \ y>1$ $\text{ }$

Upon differentiation, we obtain the pdf: $\displaystyle f_Y(y)=\beta \ y^{-(\beta+1)}=\frac{\beta}{y^{\beta+1}}, \ \ \ y>1$ $\text{ }$

The above pdf is that of a Pareto distribution with scale paramter $\alpha=1$ and shape parameter $\beta$. However, the support of this pdf is $y>1$. In order to have $y>0$ as the support, we have the following alternative version: $\text{ }$ $\displaystyle f_Y(y)=\frac{\beta}{(y+1)^{\beta+1}}, \ \ \ y>0$ $\text{ }$

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. $\text{ }$ $\displaystyle f_Y(y)=\frac{\beta \ \alpha^{\beta}}{(y+\alpha)^{\beta+1}}, \ \ \ y>0$