Beta Distribution in Actuarial Modelling

​I saw a useful way of parameterising the Beta Distribution a few weeks ago that I thought I'd write about.

The standard way to define the Beta is using the following pdf:

$$f(x) = \frac{x^{\alpha -1} {(1-x)}^{\beta -1}}{B ( \alpha, \beta )}$$

​Where $ x \in [0,1]$ and $B( \alpha, \beta ) $ is the Beta Function:

$$ B( \alpha, \beta) = \frac{ \Gamma (\alpha ) \Gamma (\beta)}{\Gamma(\alpha + \beta)}$$

​When we use this parameterisation, the first two moments are:

$$E [X] = \frac{ \alpha}{\alpha + \beta}$$

$$Var (X) = \frac{ \alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}$$

​We see that the mean and the variance of the Beta Distribution depend on both parameters - $\alpha$ and $\beta$. If we want to fit these parameters to a data set using a method of moments then we need to use the following formulas, which are quite complicated:

$$\hat{\alpha} = m \Bigg( \frac{m (1-m) }{v} - 1 \Bigg) $$
$$\hat{\beta} = (1- m) \Bigg( \frac{m (1-m) }{v} - 1 \Bigg) $$

This is not the only possible parameterisation of the Beta Distribution however. ​We can use an alternative definition where we define:

$$\gamma = \frac{ \alpha}{\alpha + \beta} $$, and $$\delta = \alpha + \beta$$

And then by construction, $E[X] = \gamma$, and we can calculate the new variance:

$$V = \frac{ \alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} = \frac{\gamma ( 1 - \gamma)}{(1-\delta)}$$.

Placing these new variables back in our pdf gives the following equation:

$$f(x) = \frac{x^{\gamma \delta -1} {(1-x)}^{\delta (1-\gamma) -1}}{B ( \gamma \delta, \delta (1-\gamma) -1 )}$$

So why would we bother to do this? Our new formula now looks more complicated to work with than the one we started with. There are however two main advantages to this new version, firstly the method of moments is much simpler to set up, our first parameter is simply the mean, and the formula for variance is easier to calculate than before. This makes using the Beta distribution much easier in a Spreadsheet. The second advantage, and in my mind the more important point, is that since we now have a strong link between the central moments and the two parameters that define the distribution we now have an easy and intuitive understand of what our parameters actually represent.
​
As I’ve written about before, rather than just sticking with the standard statistics textbook version, I’m a big fan of pushing parameterisations that are both useful and easily interpretable, The version of the Beta Distribution presented above achieves this. Furthermore it also fits nicely with the schema I've written about before (most recently in the in the post below on negative binomial distribution), in which no matter which distribution we are talking about, the first parameter of a distribution gives you information about it's mean, the second parameter  gives information about its volatility, etc. By doing this you give yourself the ability to compare distributions and sense check parameterisations at a glance.