1. An experiment was designed to compare the performances of high-speed
turbine engine made out of five different compounds. The exp eriment tested 10
bearing of each type. The time to fatigue failure are given in units of million of
cycles in my web site.
(a) Assuming that the failure time in each sample came from a Weibull distri-
bution, obtain the maximum likelihood estimates of parameters and find 95%
confidence interval for the tenth percentile of each distribution.
(b) Give the plot of five survivor functions on the same axes. carry out a
comparison of the five survivor functions. Test the equality of five survivor
functions.
(c) Investigate whether a log-normal distribution also fits the data by considering
a generalized gamma model. Compare confidence interval for tenth percentile
under a log-normal model with those in part (a)
2. The survival times of thirty-one patients with advanced melanoma treated
with combined chemotherapy, immunotherapy, and hormonal terapy are given
in my web page, where + indicate observation is censored.
(a) Make a probability plot for each of the following distributions: exponential,
Weibull, lognormal, and log-logistic distribution. Which distribution provides a
reasonable fit to the data? Estimate the parameter of the distribution chosen.
(b) Make the Cox-Snell residual plot for each of the following distributions: ex-
ponential, Weibull, lognormal, log-logistic, and generalized gamma distribution.
Which distribution provides a reasonable fit to the data?
(c) Obtain the log-likelihoods for exponential, Weibull, lognormal, and general-
ized gamma distribution. Perform the likelihood ratio test and select the best
distribution among these four distributions.
(d) Use BIC and AIC procedure to select the best distribution among the four
distributions in part (c) plus the log-logistic distribution.
(e) Compare the results obtained in parts (a)−(d). In addition, compare
the maximum likelihood estimates of the parameters with those obtained by
graphical methods.
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