Solved – Beta distribution from mean and quantile

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Suppose I'm given the mean and one quantile (e.g. the 20% quantile) of a random variable $x$, and I want to find the parameters $\alpha$ and $\beta$ of a Beta distribution that has the same mean and quantile. Is there an efficient way to do it?

Using just the mean, I know that since $\bar{x} = \frac{\alpha}{\alpha+\beta}$, we have $\beta = \frac{\alpha}{\bar{x}} – \alpha$. So we only really have one parameter to estimate. But I'm unsure how to use the quantile information to take the next step. Maybe there's something I can do with the Incomplete Beta when I know the ratio $\frac{\beta}{\alpha} = \frac{1-\bar{x}}{\bar{x}}$?

I have access to R myself, so I could use a numerical optimizer for this, but ideally I need a method that can be carried out in Excel in someone else's environment. Excel does have BETA.DIST() and BETA.INV() functions available. A look-up table would be fine, but a closed-form formula would be better if it's possible.

Best Answer

If you really have to do it with pesky Excel:

Create cells with quantile probability $p$, quantile value $q$, mean $m$.
Create a cell with some initial $\alpha$ value. Create a cell with formula $\beta=\left(\frac{1-m}{m}\right)\alpha$.
Create a cell with formula $\mathrm{abs}(q - \mathrm{beta.inv}(p, \alpha,\beta))$.
Go to "Data" > "What-If Analysis" > "Goal Seek". Choose the previous cell for item "Set cell", put $0$ in "To value", and choose the $\alpha$ cell for "By changing cell". Press "OK".
Next time: Use R! (I'm joking. I know you're an R user.)

Related Solutions

Solved – How to calculate standard error of sample quantile from normal distribution with known mean and standard deviation

This at least gives some pointers for - and a partial answer to - this question.

In the case of sample quantiles, the standard error depends on which definition of sample quantiles you actually use. I believe R, for example, includes 9 different definitions of quantiles in its quantile function.

For cases where the sample quantile is an exact order statistic, the standard error of the sample quantile follows from the standard error of that order statistic.

If a quantile is based on some weighted average of two order statistics, then the standard error can be obtained from their variances, their covariance and the weights.

As a result, confidence intervals can be formed; in the case of a quantile being an order statistic, a binomial distribution can be used to form a nonparametric interval directly from order statistics.

Solved – Tail bounds for Beta distribution

We write $I_{\epsilon}(A,B,\epsilon)$ the incomplete beta function. Thus $P(X>\epsilon) = 1-I_{\epsilon}(A,B,\epsilon)$

A simple bound is

$$\frac{2\sqrt{\epsilon}}{2\sqrt{\epsilon}+B\left(\frac{1}{2},\frac{d}{2}\right)}< I_\epsilon\left(\frac{1}{2},\frac{d}{2},\epsilon\right)<\frac{2 \sqrt{\epsilon}}{B\left(\frac{1}{2},\frac{d}{2}\right)}$$

A tighter lower bound (so upper bound for the tail) would be

$$\frac{2 \left(\sqrt{\epsilon} B\left(\frac{1}{2},\frac{d}{2}\right)+2 \epsilon\right)}{2 \sqrt{\epsilon} B\left(\frac{1}{2},\frac{d}{2}\right)+B\left(\frac{1}{2},\frac{d}{2}\right)^2+4 \epsilon}$$

Mathematica will spit them out for you with

1 - 1 / Normal[Series[1/(1 - BetaRegularized[\epsilon, 1/2, d/2]), {\epsilon, 0, n}]]

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Related Solutions

Solved – How to calculate standard error of sample quantile from normal distribution with known mean and standard deviation

Solved – Tail bounds for Beta distribution

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