Stat 567:Statistical Reliability
Eleventh Class (October 8, 1997)
Maximum Likelihood Estimation:
Using JMP's Nonlinear Fit Platform/
Univariate Delta Method
Class Objectives:
Learn to use JMP's Nonlinear platform to calculate maximum likelihood
estimates using the log-likelihood function
Learn to use the univariate Delta Method.
Homework Assignments:
Obtain the variance covariance matrix for the Normal in Example 3.2,
Page 57 of your textbook given by
from the JMP output of the Normal 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 normal fit of the log of the ball bearings data. (Given in the class notes.)
Consider the aircraft components data
of Example 2.2, Page 43. Assume an exponential distribution. Find b10 and its standard error. Calculate a 95% confidence interval
for b10.
Consider the two confidence intervals for S(.5) given in class for
the aircraft components data of Example
2.2, Page 43.
What are the assumptions leading to each one?
Which confidence interval do you prefer? Justify your answer with an
exponential plot of the data.
Why were we able to use the empirical survivor formula for the aircraft components data of Example 2.2,
Page 43 instead of the more complex Product-Limit estimator formula?
Class Outline and Main Points:
JMP's Nonlinear Fit platform loss function for the normal with no censoring
Univariate Delta Method
Motivation: the exponential mean
Derivation
Asymptotic normality
Estimation of Quantiles
Exponential case
Standard errors
Confidence intervals
Estimation of Survival probabilities
Exponential case
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?
