Class and Homework Log
Stat 567: Statistical Reliability
Ramón V. León
All homework is due on the Tuesday following the class
date on which it is assigned unless otherwise specified. Make sure to identify
all homework with the class number.
To avoid confusion homework from different classes should
be handed in distinct packages. All pages of your homework should have your
name and class number.
Classes 25 - 30 (November 24 - December
10, 1998)
Class 24 (November 19, 1998)
Class 23 (November 17, 1998)
- Read all my notes on Bayesian statistics and be ready
to ask questions in class
- Review Bayes's theorem in your theory book
Class 22 (November 12, 1998)
- Fit a propotionals hazars model to the Kevlar
49 fibers data ignoring the spool effect. Then obtain the uncensored
residuals. You are to turn in the exponential plot of these residuals.
Class 21 (November 10, 1998)
Class 20 (November 5, 1998)
- Fit model M2 that
extends the weak link model to the Single
fibers data. Your are hand in the JMP output. Then find the contribution
to the loglikelihood of model M3 that
comes from the 1 mm long fibers. You are to hand in the JMP output.
- Analyze the Tungsten carbide
alloy data. Write a report documenting your analysis. Consider the
models discussed in class. Do the appropriate diagnostic plots. Do the
appropriate residual analyses.
- The homework above is due November 19
Class 19 (November 3, 1998)
- Reproduce the last two graphs in the following handout:
Proportional Odds Model
Check
- Do a Weibull regression on the Single
fibers data with a quadratic length term using JMP's survival platform.
You are to hand in the JMP output. Do a Weibull plot of the residuals for
this model. You are to hand in this plot.
Class 18 (October 29, 1998)
Class 17 (October 27, 1998)
Class 16 (October 22, 1998)
- Analyze the Rats data using
a lognormal regression with diet treated as a categorical regressor. Check
the assumptions of the model with the lognormal plots for each diet superimposed.
Explain. Write a short report on your analyses.
Class 15 (October 20, 1998)
- Consider the following data: Kevlar
49 fibers. Use JMP's nonlinear fit platform to fit the Weibull regression
model where mu depends on both the log stress and the spool and log sigma
depend on the log stress. (This is model M4 discussed in class) You are to turn in the nonlinear fit platform
output.
- Use JMP to calculate the residuals for the model fitted
above. Then produce the box plots of residuals versus spool number and
the Weibull plots of residuals grouped by stress level.
- Do a Weibull plot of the residuals (i.e., not grouped
by stress level). Do they seem to follow a Weibull distribution? (Turn
in this Weibull plot.)
Class 14 (October 13, 1998)
- Midterm exam 2 day - no additional homework
Class 13 (October 8, 1998)
- Midterm exam 2 is on Tuesday, October 13. The exam covers
the material on likelihood methods up to PP and SP plots. This is the material
in Chapter 3 of you book. Notice that this does not include any regression
material. Good luck!
- No additional homework.
Class 12 (October 6, 1998)
- Reproduce the last JMP output in Accelerated
Life Model: Lognormal Regression Using Nonlinear Fit JMP Platform.
Using this output calculate var(sigma), cov(beta0, sigma), cov(beta1, sigma).
- Consider the insulators
data. Using the delta method on the median function of time (Approach
1) calculate a 95% confidence interval of the median life at 30 kv
- Consider the insulators
data. Using the delta method on the median function of log time (Approach
2) calculate a 95% confidence interval of the median life at 30 kv
- Consider the insulators
data. Use the Nonlinear Fit platform of JMP to calculate a likelihood
ratio 80% confidence interval for the median life at 26 kv. Notice that
you can come up with exact initial values using the information from earlier
outputs.
- Consider the insulators
data. Use the Nonlinear Fit platform of JMP to calculate a likelihood
ratio 95% confidence interval for the median life at 30 kv. Notice that
you can come up with exact initial values using the information from earlier
outputs.
Class 11 (October 1, 1998)
Class 10 (September 29, 1998)
- Calculate a likelihood ratio confidence interval for
b1 for the Weibull fit of the
Ball bearings data following the example
in Likelihood Ratio
Confidence Interval for b10: Weibull Case. (Hand in the JMP print out.)
- Calculate a likelihood ratio confidence interval for
S(53) for the Weibull fit of
the cord strengths data following the example
given in class. (Hand in the JMP print out.)
- Consider the cord strengths
data. Use a likelihood ratio test to test the hypothesis of exponential
versus Weibull. Be sure to hand in the JMP output where you maximize the
likelihoods.
- Use the tests of lognormal versus Weibull on the Aircraft components data.
Class 9 (September 24, 1998)
Class 8 (September 22, 1998)
- Midterm exam 1 on this day. No additional homework.
Class 7 (September 17, 1998)
- The first midterm exam will be on Tuesday, September
22. It will be on the material covered in class of up to the analysis of
data with multiple modes of failure. The exam will be open book and open
notes. You need a scientific calculator to take this exam.
- Verify the calculations leading to the numerical values
of estimates and standard errors on Table 3.2, Page 58 of your textbook.
- For the Ball Bearing data
with a Weibull fit calculate a 95% confidence interval for b10 using Approach 2 for calculating confidence
intervals of Weibull quantiles. That is, calcultate a 95% confidence interval
for b10 of the Gumbel fit of
log data. Then translate this cofindence interval into one for b10 of the Weibull fit of the data.
Class 6 (September 15, 1998)
- The first midterm exam will be on Tuesday, September
22. It will be on the material covered in class of up to the analysis of
data with multiple modes of failure. The exam will be open book and open
notes. You need a scientific calculator to take this exam.
- Reproduce the last JMP output in "Using
JMP's Nonlinear Platform to Create Likelihood Contours: Weibull/Gumbel
case."
- Obtain the variance covariance matrix for the Gumbel
in Example 3.2, Page 57 of your textbook given by
from the JMP output of the Gumbel fit of the log of the
ball bearings data. (Given in the class
notes.) Clearly show all formulas that you are using.
- 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.
- Write a proposal for you class project.
Check the class syllabus for more information
on the project.
Class 5 (September 10, 1998)
- The first midterm exam will be on September 22. It will
be on the material covered in class of up to the analysis of data with
multiple modes of failure. The exam will be open book and open notes. You
need a scientific calculator to take this exam.
- Analyze the appliance data supplied in class.
(You can download these data here: Appliance.
Recall that you must add a censor column to the JMP data
file since failure code zero really corresponds to removal times. Failure
code 0 should be eliminated from the Cause-Code column.)
- Write a professional
quality report. Attach a cover letter directed at management. The report
itself should be directed at the scientists or engineers who are neither
management nor statisticians but are technically oriented. The report should
have:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other things that interfere
with the smooth flow of the Data Analysis section of the report.
Hint: Use Analyzing
of Failure Data with Competing Risks to guide your analysis.
- Reproduce the JMP analyses given in the following handouts.
You are to hand in the last JMP output for each of them
- Fit a Gumbel model to the log of the Ball
bearings data using JMP's Nonlinear Fit platform
- Fit a normal model to the log of the Ball
bearings data using JMP's Nonlinear Fit platform. Is there a short
cut for calculating the values of the MLEs? Would this short cut work if
the data were right censored?
Class 4 (September 8, 1998)
- Identify the standard cummulative distribution function
G that leads to the Gumbel and normal QQ plot. Find G inverse for the Gumbel
plot.
- Use JMP to do Exponential, Weibull and lognormal plots
of the Aircraft component data of Example 2.2 on Page 43 of your textbook. Interpret the results.
You can download a text file containing
these data. (Instructions
for importing a text file into JMP)
- Let A1, A2, A3, and A4
be nested events with A1 being the smallest and A4
being the largest. Show that
P(A1)=P(A1|A2)P(A2|A3)P(A3|A4)P(A4)
What does this identity have to do with the PL estimator?
- Show that for the data in Example 2.2 on Page 43 of your
textbook the Product Limit estimator is the same as the empirical survivor
function
- Learn to use JMP in the manner shown in the following
handout: Calculation
of Kaplan Meier Product-Limit (PL) Estimator
- Analyze the rat diet data supplied in class. (You can
download these data here: Rats )
Write a professional
quality report. Attach a cover letter directed at management. The report
itself should be directed at the scientists or engineers who are neither
management nor statisticians but are technically oriented. The report should
have:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other things that interfere
with the smooth flow of the Data Analysis section of the report.
Hint: Use Comparing
Populations to guide your analysis.
- Two vendors (1 and 2) supply circuit packs used in a
system. A new system is being developed and it is necessary to choose one
of these two suppliers as the sole source on the basis of the more reliable
product. A sample of 1041 circuit packs from vendor 1 and 1245 circuit
packs from vendor 2 are put on life test and continuously monitored for
1 year, at which time the test is terminated. (These data can be downloaded
here: Vendors.) Can a definitive choice
between vendors be made on reliability considerations alone, or must other
factors be given importance since there is little to choose on the basis
of reliability?
Class 3 (September 3, 1998)
- For the ball bearing data of Example 2.1 on Page 37 of
your textbook. obtain 95% confidence intervals for S(50) and H(50)
.
- For the ball bearing data of Example 2.1 on Page 37 of
your textbook plot the empirical survivor function using the plotting points
derived on Page 40. Then plot the empirical integrated hazard. Do you think
that the underlying distribution of the data is IFR? Why?
Class 2 (September 1, 1998)
- Show that if T is a positive random variable with an
exponential distribution then:
- Show that if T measured in minutes is exponential with
hazard rate lambda, then T*=T/60 measured in hours is exponential with
hazard rate 60 time lambda.
- Show that the Weibull distribution is:
- IFR if the shape parameter is greater than 1
- DFR if the shape parameter is less than 1
- Exponential if the shape parameter is equal to 1
- Show that alpha is the 63.2 percentile of the Weibull
distribution
- Show that if T is a random variable with Weibull survivor
function
Then X=logT has a Gumbel survivor function
where
Class 1 (August 27, 1998)
Do you have something
to tell me?