Method of Moments: Beta Distribution

Given a collection of data that may fit the beta distribution, we would like to estimate the parameters that best fit the data. We illustrate the method of moments approach on this webpage. Click here to see another approach, using the maximum likelihood method.

Parameter estimates

As shown in Beta Distribution, we can estimate the sample mean and variance for the beta distribution by the population mean and variance, as follows:

Mean variance beta distribution

We treat these as equations and solve for α and β. From the first equation, we get

Beta estimation

Substituting this term for β in the second equation and then multiplying the numerator and denominator by x̄3 yields

and so

In the pure method of moments, we need to substitute t2 for s2 in the above equations.

Example

Example 1: Determine the parameter values for fitting the data in range A4:A21 of Figure 1 to a beta distribution.

We see from the right side of Figure 1 that alpha = 2.8068 and beta = 4.4941. Note too that if we calculate the mean and variance from these parameter values (cells D9 and D10), we get the sample mean and variances (cells D3 and D4).

Fit for beta distribution

Figure 1 – Fit for a Beta Distribution

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Wikipedia (2017) Beta distribution: method of moments
https://en.wikipedia.org/wiki/Beta_distribution#Method_of_moments

2 thoughts on “Method of Moments: Beta Distribution”

  1. You’ve done a huge amount of work here. And you probably need a huge manual to explain it all.
    Do you explain anywhere the parameters used in your BETA_FITM function? In the example above you have 2 parameters, the data range , and one taking the value TRUE. Not clear what the latter is referring to. When I employ the same function from your general menu using “ctrl+m,” the output gives me 2 versions, one using BETA_FITM(range, TRUE) and one with BETA_FITM(range,TRUE,TRUE). Again, I’m not clear what the TRUE/FALSE parameters refer to.

    Reply

Leave a Comment