Question

Suppose that a continuous, positive random variable T has a hazard function defined by

h(t) = h_j when a_j-1 ≤ t < a_j,   j = 1, 2, …, k,

where a₀ = 0 and a_k= ∞, a₀ < a₁ < … < a_k   and   h_j> 0, j = 1, 2, …, k.

(a) Express S(t), the survivor function of T, in terms of the h_j’s.

(b) Express f(t), the probability density function of T, in terms of the h_j’s.

(c) If you observe a random sample of size n of the form (t₁, δ₁), …, (t_n, δ_n), where δ_i = 1 if t_i is a failure time, and 0 if t_i is a right-censoring time, derive the maximum likelihood estimates of the h_j’s and, hence, the maximum likelihood estimate of S(t).

(d) What happens to the maximum likelihood estimate of S(t) as k becomes “large” and the distances a_j– a_j-1get “small”?

Answer 1