Class Objectives:

• Learn to compute Maximum Likelihood Estimators (MLEs) for the Weibull distribution
• Learn about some numerical issues in MLE calculations
• Learn to use JMP to obtain MLEs in the Weibull and lognormal case.

Homework Assignments:

• Verify that the Weibull MLEs of the ball bearing data presented in Example 3.2, Page 57 of your textbook satisfy Equations (3.8) and (3.9) on Page 55.
• Verify the variance-covariance matrix of the Weibull MLE estimates of the ball bearing data presented in Example 3.2, Page 57 of your textbook satisfy the equations on top of Page 56.
• Submit two new study questions for this or a prior lesson. Remember that these questions could end up in the exams

Class Outline and Main Points:

• Computation of MLEs for the Weibull and Gumbel distributions with right censoring
  • Solving the equations of partial derivatives equated to zero
    • Initial algebraic simplification of equations
    • Final numerical solution by method of repeated bisections
  • Computation of information matrix and variance-covariance matrix
  • Algebraic manipulations for increasing calculation precision
• Ball Bearing Data of Example 2.1, Page 37 of your textbook.
  • MLEs for a Weibull model
  • Comparison with method-of-moments and graphical estimates
  • MLEs for a lognormal model
• Using JMP to obtain MLEs

Using JMP to obtain MLEs:

To get the MLEs with JMP of the Weibull and Gumbel (Extreme-value) fits of the ball bearing data of Example 2.1, Page 37 of your textbook go to JMP's survival platform and obtain the P-L survival estimate as we have done before. Then proceed as follows:

To get MLEs for a log-normal model of the ball bearing data proceed as above but select the log-normal plot instead of the Weibull plot.

Notice that with this JMP platform we do not get the variance-covariance matrix of the estimates. Later we will show how to get this matrix using JMP's non-linear platform.

Study Questions:

• Why do we need numerical techniques to obtain MLEs?

SAS JMP file (Mac) of classroom example

• Ball Bearing data of Example 2.1, Page 37 of your textbook.

Find out how to download JMP files.

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