1. Consider experiments with the following censoring mechanism: A group of n units is observed from time 0; observation stops at the time of the rth failure or at time C, whatever occurs first. Show by direct calculation that the likelihood function is of the form L = Yn i=1 f(ti) δiS(ti+)1−δi , assuming that the units gave failure times which are i.i.d. with survivor function S(t) and p.d.f. f(t). (Hint: first define ti and δi .)

2. Suppose that T is a survival random variable with survival function S and cumulative hazard function H(t) = − log S(t). Show that H(T) ∼ exp(1).

3. Suppose that the lifetime Ti has hazard function hi(t) and that Ci is a random censoring time associated with Ti . Define λi(t) = lim ∆t→0 P(t ≤ Ti ≤ t + ∆t|Ti ≥ t, Ci ≥ t) ∆t (a) Show that if Ti is independent of Ci , hi(t) = λi(t). (b) Suppose that there exists an unobserved covariate Zi which affects both Ti and Ci , as follows: P(Ti ≥ t|Zi) = exp(−Ziθt), P(Ci ≥ t|Zi) = exp(−Ziρt), and Ti , Ci are independent, given Zi . Assume that Zi has a gamma distribution with density function g(z) = φ φ Γ(φ) z φ−1 e −φz(z > 0). Show that the joint survivor function for Ti , Ci is P(Ti ≥ t, Ci ≥ s) = 1 + 1 φ θt + 1 φ ρs−φ .

4. The lifetime of an article is thought to have an exponential distribution. Twelve such articles were selected at random and tested until nine of them failed. The nine observed failure times were 8, 14, 23, 32, 46, 57, 69, 88, 109. Assume that the data follow the exponential distribution. (a) Compute the maximum likelihood estimate of mean µ. (b) Compute the Fisher information for ˆµ. (c) Obtain a 90% confidence interval for µ by using the quantity Z = (ˆµ−µ)/se(ˆµ) where se(ˆµ) is the standard error for the estimate ˆµ.

The price is based on these factors:

Academic level

Number of pages

Urgency

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Delivering a high-quality product at a reasonable price is not enough anymore.

That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more