Q1. Consider a random censorship model in which failure time T^~ is exponential with failure rate λ and censoring time C is exponential with rate α. Let T_i = T^~_I ∧ C_i, V = _i=1∑ⁿT_i, and D be the number of failures. Show that (V, D) is sufficient for (λ, α). Show further that V and D are independent, that D is binomial with parameters n and λ(λ + α)^-1, and that 2(λ + α)V has a χ² distribution with 2D degrees of freedom. Discuss how inference on λ may be carried out.

Q2. The guaranteed exponential distribution has density function

f(t) = λe^λ(t-G),   t > G,

(a) Show that n(T₍₁₎ - G), (n - 1)(T₍₂₎ - T₍₁₎), (n - 2)(T₍₃₎ - T₍₂₎), . . . , (T_(n) - T_(n-1)) are independent exponentials with failure rate λ and hence determine the joint distribution of U and T₍₁₎.

(b) Establish methods for exact, interval estimation of λ and G. (The likelihood ratio statistic gives rise to simple pivotals for these parameters.)

Q3. Let T₁, . . . ,T_n and S₁, . . . , S_n be uncensored samples from two guaranteed exponential distributions with parameters (λ₁, G₁) and (λ₂, G₂), respectively.

(a) Outline a test of the hypothesis λ₁ = λ₂.

(b) Supposing that λ₁ = λ₂ = λ is known, develop a test of the hypothesis G1 = G2. For this purpose, show that U = T₍₁₎ - S₍₁₎ has a double exponential distribution with density

(c) Generalize this to a test of G₁ = G₂ when λ₁ = λ₂ = λ but λ is unknown. Verify that this is the likelihood ratio test of this hypothesis.

Q4. Freireich et al. (1963) present the followi.ng remission times in weeks from a clinical trial in acute leukemia:

(a) Test the hypothesis of equality of remission times in the two groups using Weibull, log-normal, and log-logistic models. Which model appears to fit the data best?

(b) Test for adequacy of an exponential model relative to the Weibull model.

(c) As a graphical check on the suitability of exponential and Weibull models, compute the Kaplan-Meier estimators F(t) of the survivor functions for the two groups. Plot log F(t) versus t and log[-logF(t)] versus log t.