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.)
Similar Posts:
- Solved – Estimation of quantile regression by hand
- Solved – Expected value of $1/x$ when $x$ follows a Beta distribution
- Solved – Beta as distribution of proportions (or as continuous Binomial)
- Solved – How to calculate the PDF of the ‘difference’ between two Beta distributions
- Solved – How to calculate a partial expected value of beta distribution (mean of a truncated beta)