Solutions for the questions (advanced survival analysis)

 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. 

Calculate your paper price
Pages (550 words)
Approximate price: -

Why Work with Us

Top Quality and Well-Researched Papers

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

Professional and Experienced Academic Writers

We have a team of professional writers with experience in academic and business writing. Many are native speakers and able to perform any task for which you need help.

Free Unlimited Revisions

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account or by contacting our support.

Prompt Delivery and 100% Money-Back-Guarantee

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. In case you cannot provide us with more time, a 100% refund is guaranteed.

Original & Confidential

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text. We also promise maximum confidentiality in all of our services.

24/7 Customer Support

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

Calculate the price of your order

Total price:

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.


Essay Writing Service

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.


Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.


Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.


Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.