# Approximating the distribution of a linear combination of beta-distributed independent random variables

This question is related with these other two questions in Cross Validated, which has been already answered:

In short, my question is this: Should I use specific results such as those collected by Gupta and Nadarajah (2004) (see also the answer by @kjetil-b-halvorsen to a previous question) to approximate the distribution of the linear combination of $n=20$ independent beta-distributed random variables, or would the CLT be accurate enough in this case? The context: statistical quality control on the production of a standard industrial setting (not NASA, I mean).

This is my concrete situation:

I have a sequence $X_1, X_2, X_3, \dots$ of independent random variables that can be assumed to follow a beta distribution, each of them with their respective distribution parameters, not necessarily equal — that is:

$$X_i \sim \mathrm{Beta}(a_i,b_i) \text{,} \quad \forall\; i \text{.}$$

Actually, all $X_i$'s should have the same distribution. I mean, theoretically speaking, there is an underlying distribution $\mathrm{Beta}(a,b)$ which all the $X_i$'s should come from, but the process is not under statistical control.

I am interested in approximately determining the distribution of the average of $n$ of those $X_i$'s. Without loss of generality, I would like to approximate the distribution of

$$Y = \frac{1}{n}\sum_{i=1}^n{X_i} \text{.}$$

An approach based on concrete data is possible (I mean, calculating concrete values for $Y$ from concrete values for the sequence of $X_i$'s and try to fit a distribution) and will be done. But I am also interested in connecting the distribution of $Y$ with the distribution of the $X_i$'s in a more theoretical way, so that we be able to deduce things about $Y$ basing on what happens to the $X_i$'s.

Using the Lindeberg-Feller CLT (see https://stats.stackexchange.com/a/156464/44075), I could state —if I am not wrong— that $Y$ is approximately distributed as a normal variable with mean $\mu_Y$ and standard deviation $\sigma_Y$, where $\mu_Y$ can be estimated as the mean of a sample of $X_i$'s and $\sigma_Y$ can be estimated as the sample (quasi)standard deviation of the $X_i$'s divided by $\sqrt{n}$.

On the other hand, Johannesson and Giri (1995), who are cited by Gupta and Nadarajah (2004), provide two ways to approximate $Y$ using a beta distribution. The more complex of them says that $Y$ is approximately equal to $\rho Z/\gamma$, where $Z$ is a standard beta random variable with parameters $g$ and $h$, and where $\rho$, $\gamma$, $g$ and $h$ can be determined using explicit equations that can be translated to be estimated from a sample of $X_i$'s.

So, which of the approaches should I use? The normal approximation or the beta one?

As I said above, in my concrete case, the value of $n$ is $20$ or so.

EDIT:

I am interested in this matter because I was warned about the fact that the convergence rate of $Y$ to a normal distribution (when $n$ tends to infinity) is not stated by the Lindeberg-Feller CLT.

* I don't mean this comment as a general one, just in respect of beta variates. For example, if the skewness $\gamma_1$ of a beta variate is small the kurtosis is bounded above and below by $1 +$ a multiple of $\gamma_1^2$ (where both multiples are small), and I believe the absolute third moment of a standardized variate should be smaller than the fourth moment. Those two things together suggest a small third moment implies a small absolute third moment.
• I understand your answer, I think. In my case, the beta-distributed variables I am considering are positive-skewed (they have to do with proportions that are kind of between 5 and 10%, in average). I should plot my real data in order to see if $Y$ can be considered to be approximately normal or not. Thank you. For now, I leave the question as unanswered to get more contributions, maybe with different approaches. Jun 17, 2015 at 22:43