A limiting case of the transformed beta distribution

We will prove that transformed gamma distribution is a limiting case of transformed beta distribution with $\theta \to \infty, \alpha \to \infty, \theta/\alpha^{1/\gamma} \to \xi$.

First of all, what's a transformed beta distribution.

Let's consider the beta distribution:

$$f(x) = \frac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)}x^{a-1}(1-x)^{b-1}$$

First let $x = \frac{1}{1+u}$, then $1-x= \frac{u}{1+u}$ and we will have $dx = -\frac{1}{(1+u)^2}\, du$, and our interval will be $0\le x\le1$ transform into $\infty$ to $0$. So we will have

$$f(u) = \frac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)}\left(\frac{1}{1+u}\right)^{a-1}\left(\frac{u}{1+u}\right)^{b-1}\frac{1}{(1+u)^2}$$

We will then let $u = \left(\frac{y}{\theta}\right)^{\gamma}$, so $du = \gamma\left(\frac{y}{\theta}\right)^{\gamma-1}\frac{1}{\theta}$ and $$\frac{1}{1+u} = \frac{1}{1+\left(\frac{y}{\theta}\right)^{\gamma}} = \frac{\theta^{\gamma}}{\theta^{\gamma} + y^{\gamma}}$$

and

$$\frac{u}{1+u} = \frac{\theta^{\gamma}\left(\frac{y}{\theta}\right)^{\gamma}}{\theta^{\gamma}+ y^{\gamma}} = \frac{y^{\gamma}}{\theta^{\gamma}+y^{\gamma}}$$

so overall, we are having

$$f(y) = \frac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)}\left(\frac{\theta^{\gamma}}{\theta^{\gamma} + y^{\gamma}}\right)^{a+1}\left(\frac{y^{\gamma}}{\theta^{\gamma}+y^{\gamma}}\right)^{b-1}\gamma y^{\gamma-1}\theta^{-\gamma}$$

$$=\frac{\Gamma(a+b)}{\Gamma(a) \Gamma(b)}\theta^{a\gamma}y^{\gamma b-1}\frac{1}{(\theta^{\gamma}+y^{\gamma})^{a+b}} \gamma $$

$$=\frac{\Gamma(a+b)\gamma }{\Gamma(a) \Gamma(b)}y^{\gamma b-1}\frac{\theta^{-b\gamma}}{(1+\left(\frac{y}{\theta}\right)^{\gamma})^{a+b}} $$

$$= \frac{\Gamma(a+b)\gamma }{\Gamma(a) \Gamma(b) y}\frac{\left(\frac{y}{\theta}\right)^{\gamma b}}{(1+\left(\frac{y}{\theta}\right)^{\gamma})^{a+b}}$$

And the above is the transformed beta distribution with regarding to $y$ for $y \ge 0$.

We will prove that transformed gamma distribution is a limiting case of transformed beta distribution with $\theta \to \infty, a \to \infty, \theta/a^{1/\gamma} \to \xi$.

Now we have the definition of transformed beta distribution, we can consider our problem.

$$f(x) = \frac{\Gamma(a+b)\gamma }{\Gamma(a) \Gamma(b) x}\frac{\left(\frac{x}{\theta}\right)^{\gamma b}}{(1+\left(\frac{x}{\theta}\right)^{\gamma})^{a+b}}$$

By using Stirling's approximation, we know that $\Gamma(x) \to \sqrt{\frac{2\pi}{x}} \left(\frac{x}{e} \right)^{x} $ as $x$ goes to infinity. So we will have

$$\lim_{a \to \infty}f(x) = \lim_{a \to \infty}\frac{\sqrt{\frac{2\pi}{a+b}} \left(\frac{a+b}{e} \right)^{a+b}}{\sqrt{\frac{2\pi}{a}} \left(\frac{a}{e} \right)^{a} \Gamma(b)}\frac{\gamma}{x} \frac{\left(\frac{x}{\theta}\right)^{\gamma b}}{(1+\left(\frac{x}{\theta}\right)^{\gamma})^{a+b}}$$

$$=\frac{\gamma}{x\Gamma(b)} \lim_{a \to \infty} \left(\frac{1}{1+\left(\frac{x}{\theta}\right)^{\gamma}}\right)^b \left(\frac{x}{\theta}\right)^{\gamma b}\frac{ \left(\frac{a+b}{e} \right)^{a+b}}{ \left(\frac{a}{e} \right)^{a} }{(1+\left(\frac{x}{\theta}\right)^{\gamma})^{-a}}$$

$$= \frac{\gamma}{x\Gamma(b)}\lim_{a \to \infty} \left(\frac{1}{1+\left(\frac{x}{\theta}\right)^{\gamma}}\right)^b \left(\frac{x}{\theta}\right)^{\gamma b} \left(1+\frac{b}{a} \right)^{a}\left(\frac{a+b}{e} \right)^b{\left(1+\left(\frac{x}{\theta}\right)^{\gamma}\right)^{-a}}$$

So we will have $$=\frac{\gamma}{x\Gamma(b)}\lim_{a \to \infty} \left(\frac{x}{\theta}\right)^{\gamma b} (a+b)^b{\left(1+\left(\frac{x}{\theta}\right)^{\gamma}\right)^{-a-b}}$$

At the same time we take $\theta \to \infty$, we also note that $\lim_{\theta \to \infty }\theta = \lim_{a \to \infty}\xi a^{1/\gamma} $, which means $\lim_{\theta \to \infty }\theta^{\gamma} = \lim_{a \to \infty}\xi^{\gamma} a$ , so we will have

$$\lim_{\theta \to\infty }\lim_{a \to \infty} f(x) = \frac{\gamma}{x\Gamma(b)}\lim_{\theta \to \infty, a\to \infty} x^{\gamma b} (\theta^{\gamma})^{-b} (a+b)^b{\left(1+x^{\gamma} \theta^{-\gamma}\right)^{-a-b}}$$

$$\lim_{\theta \to\infty , a \to \infty} f(x) = \frac{\gamma}{x\Gamma(b)}\lim_{\theta \to \infty, a\to \infty} x^{\gamma b} (\xi^{\gamma}a)^{-b} (a+b)^b{\left(1+x^{\gamma} \xi^{-\gamma}a^{-1}\right)^{-a-b}}$$

$$= \frac{\gamma}{x\Gamma(b)}\lim_{ a\to \infty} x^{\gamma b} \xi^{-b\gamma}a^{-b} (a+b)^b{\left(1+x^{\gamma} \xi^{-\gamma}a^{-1}\right)^{-a-b}}$$

$$= \frac{\gamma x^{\gamma b} \xi^{-b\gamma}}{x\Gamma(b)}\lim_{a\to \infty} {\left(1+x^{\gamma} \xi^{-\gamma}a^{-1}\right)^{-a-b}}$$

$$=\frac{\xi^{-\gamma b}}{\Gamma(b)} \gamma x^{\gamma b-1} e^{-x^{\gamma }\xi^{-\gamma}}$$

$$=\frac{\gamma}{\xi^{\gamma b}\Gamma(b)} x^{\gamma b-1} \exp\left({-\left(\frac{x}{\xi}\right)^{\gamma}}\right) $$

which is a transformed beta distribution.

A heavy-tailed distribution refers to a probability distribution that has a higher probability of extreme or rare events occurring compared to a distribution with lighter tails. In other words, heavy-tailed distributions have tails that decay more slowly, allowing for the occurrence of extreme or rare events with higher probabilities. The term "heavy-tailed" signifies that the tails of the distribution contain a significant amount of probability mass. This indicates that the distribution has a greater propensity for extreme values or outliers compared to distributions with lighter tails, where extreme events occur with lower probabilities. The heavy-tailed property has important implications in various fields, including finance, insurance, and risk management. It suggests that rare or extreme events, such as financial market crashes or catastrophic losses, can occur more frequently and have a more significant impact than what traditional models assuming lighter-tailed distributions would predict. Understanding and modeling heavy-tailed distributions are crucial in accurately assessing risks associated with extreme events and developing appropriate risk management strategies.

In the field of actuarial science, there are two heavily tailed distributions that are frequently considered. They are the following:

(1) Inverse Weibull distribution. For modeling the maximum observation from a random sample: The inverse Weibull distribution (also known as the Fréchet distribution) is a suitable choice. The maximum observation from a random sample, especially from a heavy-tailed distribution, tends to exhibit heavy tails. Hence, the inverse Weibull distribution provides a good model in situations where the goal is to understand the maximum potential claim or extreme values.

(2) Pareto distribution. For modeling excess losses with high deductibles: The Pareto distribution is often a suitable model. In insurance scenarios involving high deductibles, such as reinsurance contracts, the interest primarily lies in modeling large losses rather than moderate losses. The Pareto distribution, which represents heavy-tailed behavior, serves as a reliable choice for capturing extreme losses in such situations.