in R markdown,

MLE with Survival Data

The Weibull model can be used to model time xx until an event, such as the time until a battery burns out or a patient's cancer returns.

f(x)=k(x)k1e(x/)k, where 0x<), the scale 0<<, the shape 0

Consider the following (simulated) months until various batteries of the same type burn out:

Ex3 <- c(2.228, 2.051, 1.683, 3.285, 1.219, 2.879, 2.976, 2.112, 2.357, 2.425, 1.255, 2.562, 0.829, 2.581, 2.340, 3.043, 0.684, 1.810, 2.529, 0.700)

Assume we know that the time to burnout is Weibull, and that the shape k=3.

(a) We want to estimate the scale parameter . What is the log-likelihood for our sample?

(b) Plot the likelihood function L() as a function of .

(c) What is the maximum log-likelihood estimate of ?