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STAT 702, Fall 2022

Homework Assignments

Homework 2

Problem 1. (30 points)  Let T1 , . . . , Tn  be a random sample from Weibull(λ, γ) (see the formula of its p.d.f.  given in Problem 1 of Homework 1), and let (V1 , δ 1 ), . . . , (Vn , δn ) be the actually observed right censored sample with Vi  = min{Ti , Ci }, δi  = I{Ti  < Ci }, where the right censoring variable Ci is independent of Ti .

(a) Show that for any given value of γ > 0, the MLE for λ is given by = (∆n / Viγ )1/γ , where ∆n  = δi .

(b) Show that a critical point as a candidate of the MLE for γ should be a solution of equation:

0 = + i1 δi ln Vi _ ∆n .

(c) Show that if there are at least two distinct uncensored Vi’s and each Vi   > 0, estimating equation in part (b) always has a unique solution > 0.

Problem 2. (10 points) Let T1 , . . . , Tn  be a random sample from lifetime d.f.  FT , and let (V1 , δ 1 ), . . . , (Vn , δn ) be the actually observed doubly censored sample with

where (Ci , Di ) is independent of Ti . Derive the likelihood function for FT .

Problem 3. (10 points) Let T1 , . . . , Tn  be a random sample from lifetime d.f.  FT , and let (V1 , δ 1 ), . . . , (Vn , δn ) be the actually observed C益卜卜é﹔t ←tat益s sample with δi  = I{Ti  < Vi }, where Vi is independent of Ti . Derive the likelihood function for FT .

Problem 4. (30 points) In a study on time T (in weeks) to remission of leukemia, a right censored sample of 10 patients was recorded as

(6, 0), (13, 1), (20, 0), (6, 1), (10, 1), (32, 0), (16, 1), (22, 1), (11, 0), (7, 1).

(a) Compute the Kaplan-Meier estimator KM  for FT .

(b) Give a point-estimator for the mean µT  of T, and compute its value using above data.

(c) Give a point-estimator for the variance σT(2)  of T, and compute its value using above data.