This material introduces the basic theory of maximum likelihood estimation by discussing the likelihood function, the log likelihood function, and maximizing these functions using calculus. Several exercises ask students to derive certain estimators, while others have students compare the behavior of those estimators with other possibilities through the use of various JAVA applets. The applets use the same control features: the sliders set the parameter values, the Stop # drop down menu sets the number of samples taken, the Update # drop down menu sets how often the graph and tables update during the experiment, the single arrow takes one sample, the double arrow runs the full experiment, the square stops the experiment, and the back arrow resets the applet. This page is one lesson from the Virtual Laboratories in Statistics.
Understand maximum likelihood and its importance. Derive the maximum likelihood estimator. Compare this estimator with others using the mean square error.
Target Student Population:
This material would be most appropriate for graduates and advanced undergraduates in statistics.
Prerequisite Knowledge or Skills:
Calculus through partial differentiation. Probability distribution functions.
Type of Material:
This web page is a written lesson with 32 associated exercises, four of which use JAVA applets.
A teacher could show the applets or work the exercises in this material in class or assign it as outside reading or homework, with students turning in their answers to the exercises.
This material was designed to be used with Mozilla Firefox with the MathML fonts installed and Java plug-in. However, it will work with Internet Explorer with the MathPlayer and Java plug-ins.
Evaluation and Observation
This material completely and concisely describes the process of maximum likelihood estimation. Every reference to a probability distribution contains a link to the lesson on that distribution. The exercises lead the students to further understanding.
The applets demonstrate the relative performance of the maximum likelihood estimators compared to other estimators, but no applets demonstrate the idea behind maximum likelihood estimation. Also, no solutions are given for the exercises.
Potential Effectiveness as a Teaching Tool
This material is a complete lesson from introduction to assignment, effectively introducing the idea of maximum likelihood. Users can click on the formulas given to see them in larger print. The applets provide the simulation of running an experiment numerous times to grasp the idea of comparing estimators. The text on the main page describes what the student should be looking for when using the applets.
The web page itself is all text and does not contain any graphs to make understanding easier, and there are no solutions to the exercises.
Using the applet before reading the text on the main page will prevent students from knowing what to look for. The statistics in the applets are called U, V, and W instead of being labeled by their common notation.
Ease of Use for Both Students and Faculty
The applets have similar controls, making it easy to progress from one to the next. Each has a short description and general instructions provided as well.
Users can click on blue words or phrases in order to get more information on these topics as well as the formulas to see them in larger print.
Other Issues and Comments:
These notes, applets, and exercises are a useful introduction to maximum likelihood estimation. However, students may be intimidated by the text, math formulas, and lack of graphics on the main page. Users should also be aware of the technical requirements and make sure they have the proper plug-ins. Because this material is part of the "Virtual Laboratory in Probability and Statistics" users can easily see how it is connected to other related topics.