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.