Fitting Beta Distribution Parameters via MLE

Basic Concepts

We show how to estimate the parameters of the beta distribution using the maximum likelihood approach. From the pdf of the beta distribution (see Beta Distribution), it is easy to see that the log-likelihood function is

Log-likelihood Beta Distribution

We now define the following:

g1 beta distribution

g2 beta distribution

G beta distribution

where ψ and ψ1 are the digamma and trigamma functions, as defined in Fitting Gamma Distribution using MLE.

Newton’s Method

We can now use Newton’s Method to estimate the beta distribution parameters using the following iteration:

Initial guess beta distribution

Newton's Method beta distribution

G matrix Newton's method

where all these terms are evaluated at αk and βk.

Examples Workbook

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

Reference

Wikipedia (2017) Beta distribution
https://en.wikipedia.org/wiki/Beta_distribution

2 thoughts on “Fitting Beta Distribution Parameters via MLE”

  1. Thanks a lot for the explanation and the spreadsheet. I created a copy in Numbers on the Mac.

    Could you post a screenshot of the Excel sheet. That would allow me to check whether my implementation gives the correct results,

    tnx,

    Arnaud

    Reply
    • Hello Arnaud,
      If you click on the Download button towards the bottom of the webpage, you can download the spreadsheet with the example. You can then take a screenshot of the example if you like.
      Charles

      Reply

Leave a Comment