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 Ti = T~i Λ Ci, V = i=1∑nTi, and D he 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) = λeλ(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.)
(c) Apply these results to the group 1 data of Table. For this purpose, omit the censored data points.
Q3. Let T1, . . . ,Tn and S1, . . . ,Sm 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:
|
Placebo:
|
1,
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1,
|
2,
|
2,
|
3,
|
4,
|
4,
|
5,
|
5,
|
8,
|
8,
|
|
8,
|
8,
|
11,
|
11,
|
12,
|
12,
|
17,
|
22,
|
23
|
|
|
|
6-MP:
|
6,
|
6,
|
6,
|
7,
|
10,
|
13,
|
6,
|
22,
|
23,
|
6+,
|
9+,
|
|
10+,
|
11+,
|
17+,
|
9+,
|
20+,
|
25+,
|
32+,
|
32+,
|
34+,
|
35+
|
|
(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.