Description
1. The log-logistic distribution has survival function
S(t) = 1
1 + λtα
where λ, α > 0 and t ≥ 0. Here, the distribution of the logarithm of the
failure times is logistic with mean µ and scale parameter σ, and α = 1/σ,
λ = exp(−µ/σ)
(a) Find the hazard rate for this distribution. For what values of α is the
hazard function monotone. When it is monotone, is it increasing or
decreasing?
(b) When the hazard function is not monotone, where does it change slope?
Is it decreasing to increasing or increasing to decreasing?
(c) Describe a practical situation where the hazard function might change
slopes in this manner.
2. Do Exercise 2.18 from Klein and Moeschberger.
3. Using the data from Exercise 3.6 in Klein and Moeschberger,
(a) Suppose that time from the start of the study to relapse is assumed to be
exponential with parameter λ1 and time from start of the study to death
is assumed to be exponential with parameter λ2. Find the likelihoods for
λ1 and λ2 and find the MLE for each of these parameters.
(b) Suppose that we could not observe the death times of patients who had
not relapsed. Find the estimate of λ2 from the truncated sample and
discuss the difference.
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4. Suppose that the failure distribution of a particular type of disk drive is exponential with a mean time between failures (MTBF) of 4 years (so that
λ = 0.25). In this model we suppose that after each failure, the time to the
next failure is still exponential with the same parameter λ.
This means that
the failures form a Poisson process with the mean number of failures per year
of λ = 0.25. If we have n disks on test, then the failure process is still Poisson
with MTBF 1/(nλ) and mean number of failures per year of nλ.
Suppose that we put n disks on test and need to have a 95% chance of at
least 30 failures in the first year to have sufficient accuracy in estimating λ.
How many disks need to be put on test? We can attack this either using the
Poisson distribution (qpoiss, ppoiss in R) or by using the gamma distribution
(qgamma, pgamma).
For the Poisson calculation, you need to compute the mean
number of failures in the first year as a function of n, and then you can use
qpoiss with lower=F and trial and error in choosing n until you find the
minimum value of n that results in 30 as the point with upper tail equal to
0.95.
For the gamma calculation, the waiting time to the kth failure with
exponential waiting times with parameter λ0 has rate λ0 and shape k.
You
can use pgamma to find the least value of n that results in a time to 30 failures
of one year that has a probability of 0.95 or greater. Do this calculation both
ways until you get the same answer from both methods.
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