Stat 567:Statistical Reliability
Eighth Class (September 24, 1997)

Introduction to Maximum Likelihood Estimation

Class Objectives:

• Review the theory of maximum likelihood estimation
• Learn to derive the likelihood function when there is censoring involved
• Learn to compute Maximum Likelihood Estimators (MLEs) in simple cases of reliability interest

Homework Assignments:

• Read Chapter 3 of your textbook
• Derive the likelihood function for the following data:
  • Left censoring times: 2
  • Right censoring times: 7, 7, 11
  • Failure times: 3, 5.5

(Homework due Wednesday, October 1)

Recall that you have a Midterm Exam Monday, September 29

Class Outline and Main Points:

• Likelihood function
  • General definition
  • Failures are independent - and there is:
    • no censoring
    • right censoring
    • right and interval censoring
  • Log-likelihood function
• Maximum likelihood estimation of functions of model parameters
  • Invariance property
  • Quantiles
• Asymptotic theory of maximum likelihood estimation
  • For "large samples" and mild regularity conditions MLEs are approximately normal, approximately unbiased, and have the smallest attainable variance among estimators
  • Information matrix
  • Variance-Covariance matrix
• Exponential distribution with right censoring
  • Estimators for the hazard rate and mean time to failure
  • Total time on test (TTT)

Study Questions:

• Why are maximum likelihood estimates reasonable?
• Why do we usually maximize the log-likelihood function instead of the likelihood function?
• Why is the total time on test (TTT) a useful concept?

Do you have something to tell me?