Stat 567:Statistical Reliability
Twelves Class (October 13, 1997)

General Form of the Delta Method

Class Objectives:

• Learn to use the delta method to obtain standard errors and confidence intervals for estimated reliability figures of merit
• Learn to calculate standard errors and confidence intervals for Weibull and lognormal quantiles and survival probabilities.

Homework Assignments:

• Verify the calculations leading to the numerical values of estimates and standard errors on Table 3.2, Page 58 of your textbook.
• Estimate the lower 5% percentile and its standard error for the data used in Table 3.2, Page 58. Assume a Weibull distribution
• Assuming a Weibull model for the data of Example 2.1 and 3.2 (and of Table 3.2) find the MLE of S(60) and its standard error. Compare your results with those you get without the Weibull assumption using the empirical survivor function.
• For each of the distributions listed below what functions of the parameters are used to calculate the standard errors of quantiles and survival probabilities via the Delta method:
  • Weibull
  • Gumbel
  • Exponential
  • Lognormal
  • Normal
• Rewrite the Appliance data report for Wednesday, October 22. 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 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:

• Multivariate Delta Method
  • Bivariate case
  • Trivariate case
  • Illustration: Weibull and lognormal case
  • Asymptotic normality
• Estimation of Quantiles
  • Standard errors
  • Confidence intervals
  • Weibull, Gumbel, and Lognormal cases
  • b10
• Estimation of Survival probabilities
  • Standard errors
• Confidence intervals

Study Questions:

• Why is it important to be able to calculate standard errors and confidence intervals for estimated quantiles?
• Why is it important to be able to calculate standard errors and confidence intervals for survival probabilities?
• How do you use the Delta Method to calculate standard errors of quantiles and survival probabilities?
• In the Weibull case, why is it usually better from a customer's point of view to provide confidence intervals for quantiles and survival probabilities than for the Weibull parameters?
• Why is b10 frequently easier to estimate with the data of a life test than the median life?
• How does knowledge of the Variance-Covariance matrix helps in interpreting the results of a reliability statistical analysis to a client?

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