Class Objectives:

• Learn to construct test and confidence regions based on the variance-covariance matrix
• Compare the results obtained using this method with the results obtained using likelihood ratio methods
• Compare the results of both approaches and understand which results are preferable

Homework Assignments:

• Read the notes and examples carefully. The midterm exam is fast approaching and you need the results of this sections to understand most of the rest of this course and many other areas of statistical application.
• Review the instruction on writing class reports. Then read carefully Anne Freeman's rat study report handed in class. Try to understand what made hers a professional quality report but also look for areas where it can be improved. Put her report aside and don't look at it anymore. Reanalyze the rat study data and rewrite your rat study report. I will scan your report to see if it is too similar to Anne's report. If it is - it will affect your grade very negatively. Remember that the goal is for you to learn to write a data analysis report as a professional statistician would. And for you to have something to show as an example of your writing to a prospective employers. Hint: A good report is like good short story - full of substance and with nothing extraneous. This assignment is due on Monday, October 14.

Class Outline and Main Points:

• Review of rat report homework
  • Anne Freeman's report as an example
• Likelihood methods based on the Variance-Covariance matrix V
  • Test of hypothesis
  • Confidence regions
  • Example: scale parameter of the Gumbel
    • Confidence intervals are standard error confidence intervals
    • The tests based on LR and V reject at different levels of significance
    • Shape of confidence intervals based on LR and V
• Remarks
  • Both LR and V methods frequently give similar results, but large discrepancies between the results of the two methods are possible
  • Confidence regions based on V are ellipsoids
  • Generally, prefer the results of the Likelihood Ratio Method
    • Results are invariant under reparametrization
    • Shape of the confidence region is dictated by the data by way of the likelihood function
    • Confidence regions are the smallest area posterior credibility regions when the prior is flat
    • No negative values in confidence region
• Example: Exponential distribution example
  • Confidence interval based on V for the mean
  • Confidence interval of the hazard rate obtained by taking reciprocals in the confidence interval for the mean
  • LR confidence interval for the hazard rate
  • Exact confidence interval under type I censoring
  • Closest to exact confidence interval is the LR confidence interval
    • Dependency of the exact confidence interval on the stopping rule of the life test make this result less convincing than it could be.
    • Dependency of frequentist confidence intervals on the stopping rule of the life test is a strong argument for the Bayesian point of view
  • Calculation of the exact confidence interval under type one censoring

Study Questions:

• Why are likelihood ratio results generally preferable to results obtained using the variance-covariance matrix?
• Why do we sometimes use likelihood methods based on the variance-covariance matrix instead of using likelihood methods based on the likelihood ratio?
• What do we mean by invariance under reparametrizations? Why is this a desirable property?

SAS JMP files (Mac) of classroom examples and homework

• Data on the number of revolutions to failure for each of 23 ball bearings. (Example 2.1, Page 37 of your textbook)

• Data on the life test done on thirteen aircraft components subject to Type II censoring. (Example 2.2, Page 43). Data used in the JMP analysis above.

Find out how to download a JMP file

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