Stat 567:Statistical Reliability
Thirteenth Class (October 15, 1997)

Likelihood Ratio Confidence Intervals and Tests

Class Objectives:

• Learn to construct tests and confidence intervals based on likelihood ratios
• Contrast confidence intervals based on standard errors to confidence intervals based on likelihood ratios
• Understand the logic behind likelihood ratio methods

Homework Assignments:

• Reproduce the JMP output for the exponential fit of the aircraft component data. (Given in today's class notes.)
• Reproduce the JMP output of the Gumbel fit of the log of the ball bearings data. (Given in the class notes.)
• Reproduce the JMP output of the Gumbel fit of the log of the ball bearings data with sigma locked at the value one. (Given in the class notes.)
• Read the material in the notes. The likelihood ratio method will open a whole new world of statistical application outside the traditional linear model and normal distribution framework. You need to understand the logic behind the likelihood ratio method to be able to use this method correctly.

Class Outline and Main Points:

• Relationship of the likelihood function to confidence interval based on standard error: exponential case
  • Calculation of confidence interval based on standard error
• Relationship of the likelihood function to confidence interval based on likelihood ratio: exponential case
  • Calculation of likelihood ratio confidence interval
• Applications of likelihood ratio methods
  • Test for exponential versus Weibull
  • Confidence interval for exponential
  • Confidence region for both Gumbel parameters
  • Confidence interval for Gumbel scale parameter
• Likelihood ratio methods: general results
  • Diagrams illustrating main results
  • Tests
  • Confidence regions
• Bottom line.
  • Likelihood ration methods compare the maximum of the log-likelihood to its value at various parameter settings
• The likelihood ratio test plays the role of the F test in many area of application where normality cannot be assumed

Study Questions:

• Why do you need to understand understand this "theory" to be a good applied statistician?
• Why do we need computers to be able to apply likelihood ratio methods to many applied problems?
• What do we mean when we say that likelihood ratio tests play a similar role to F tests in many areas of application where the underlying distributions are not normal?

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