Stat 567:Statistical Reliability
Tenth Class (October 6, 1997)

Maximum Likelihood Estimation:
Using JMP's Nonlinear Fit Platform

Class Objectives:

• Learn to use JMP's Nonlinear platform to calculate maximum likelihood estimates using the log-likelihood function

Homework Assignments:

• Use JMP's Nonlinear Fit platform to fit an exponential distribution to the Aircraft components data. What is the value that JMP gives you for the MLE of lambda? How does this value compare with the number of failures divided by the total time on test? What 95% confidence interval does JMP gives you for lambda? How does this confidence interval compare with the one that one would obtain using the standard error of the maximum likelihood estimate found in Example 3.1, Page 56 of your textbook? (This standard error can also be obtained from the JMP output.)
• Obtain the variance covariance matrix for the Gumbel in Example 3.2, Page 57 of your textbook given by

from the JMP output of the Gumbel fit of the log of the ball bearings data. (Given in the class notes.) Clearly show all formulas that you are using.

• Reproduce the JMP output of the Gumbel fit of the log of the ball bearings data. (Given in the class notes.)
• Verify that the values:

in Problem 3.2, Page 57 of your textbook satisfy Equations (3.8) and (3.9) on Page 55 of your textbook.

• Verify that the variance covariance matrix for the Gumbel in Example 3.2, Page 57 of your textbook given by

can be obtained by using the three equations on top of Page 56 of your textbook and the formula:

• Remember that you have to write a proposal for the final project. What experiment are you going to do? Who are you going work with? (This homework was due Monday, October 6, 1997.)
• Rewrite the rat diet report for Wednesday, October 15. Read the revised professional quality report guidelines for suggestions on improving your report. Recall that you are to attach a cover letter directed at management. Also recall that the report itself should be directed at the scientists or engineers who are neither management nor statisticians but are technically oriented. The report should have the following sections:
  • Summary and Conclusions
  • Data Analysis
  • Appendix with the raw data and other material that interferes with the smooth flow of the Data Analysis section of the report.

Class Outline and Main Points:

• How censoring codes can be used to write the likelihood function when one has right censored data
• To use JMP's Nonlinear platform to obtain MLEs:
  • One programs the kernel (summand) of the negative log-likelihood in a new JMP table column
  • One uses the parameter feature of the calculator to denote the parameter in the formula
  • Initial estimate are obtained by using the graphical estimates.
• In the Weibull case one can use Equations (3.8) and (3.9) on Page 55 of your textbook to calculated the MLEs for the Gumbel distribution of log-data. The method of repeated bisections can be used in conjunction with Equation (3.9) to obtain the MLE for sigma. Once this estimate is obtained one can find the MLE of mu using Equation (3.8). Notice that once these Gumbel MLEs are obtained the MLEs for the Weibull of the data can be obtained by using the relationship of the Weibull parameters and the Gumbel parameters.

Study Questions:

• How is the Variance-Covariance matrix related to the correlation matrix in JMP's Nonlinear platform output? What formulas do you use to obtain one matrix from the other?
• Why must one use numerical methods to find the MLEs in the Weibull case?

Do you have something to tell me?