2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1-2), for x=1,2,..., with 0 << 1. It is subject to censoring which occurs at random time C, independent of T. Denote by 7 the observed exit time and by 8 exit indicator defined respectively by T= min{T, C} and 8 = 1(sc). We are interested in estimating the parameter from independent random sample (T¡, di), i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals. Subject ID coincides with 1 T 1 1 2 3 5 Table 1: (Exit indicator d = 1 if not censored and 8 = 0 if censored.) 234 in 2 3 4 Observed exit time 5 Exit indicator 8 1 0 1 0 1 (a) Show that under independent censoring, the mortality rate under censoring H₂(x) = P(t = x₂6 = 1|T ≥ x), μ(x) = P(T = x|T > x) = λ. (b) Write the likelihood function of independent pair observations (T¡, 6;), i = 1, ..., N. (c) Find the maximum likelihood estimator of the parameter 1. Show your working. Use the dataset in Table 1 to get the value of the estimator.
2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1-2), for x=1,2,..., with 0 << 1. It is subject to censoring which occurs at random time C, independent of T. Denote by 7 the observed exit time and by 8 exit indicator defined respectively by T= min{T, C} and 8 = 1(sc). We are interested in estimating the parameter from independent random sample (T¡, di), i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals. Subject ID coincides with 1 T 1 1 2 3 5 Table 1: (Exit indicator d = 1 if not censored and 8 = 0 if censored.) 234 in 2 3 4 Observed exit time 5 Exit indicator 8 1 0 1 0 1 (a) Show that under independent censoring, the mortality rate under censoring H₂(x) = P(t = x₂6 = 1|T ≥ x), μ(x) = P(T = x|T > x) = λ. (b) Write the likelihood function of independent pair observations (T¡, 6;), i = 1, ..., N. (c) Find the maximum likelihood estimator of the parameter 1. Show your working. Use the dataset in Table 1 to get the value of the estimator.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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Transcribed Image Text:2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution
P(T = x) = (1-2)-¹, for x = 1,2,...,
with 0 << 1. It is subject to censoring which occurs at random time C, independent of
T. Denote by 7 the observed exit time and by 8 exit indicator defined respectively by
T = min{T, C} and 8 = 1(7<c).
We are interested in estimating the parameter from independent random sample (7,, di),
i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals.
Subject
ID
1
2
3
coincides with
1
2
3
5
Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.)
4
5
Observed
exit time
Exit
indicator
8
(a) Show that under independent censoring, the mortality rate under censoring
H₂(x) = P(T= x,6 = 1|T≥ x),
0
1
J(X) =
μ(x) = P(T = x|T> x) = λ.
(b) Write the likelihood function of independent pair observations (T;, di), i = 1, ..., n.
(c) Find the maximum likelihood estimator of the parameter 1. Show your working.
Use the dataset in Table 1 to get the value of the estimator .
(d) Show that the observed Fisher information J() is given by
- — £ ₁ + 2 -
di
(1-2)²
FI
(Ti-1).
(e) Calculate an estimate Var() of the variance of λ. Use the dataset.
(f) Give the 95% confidence interval for 1.
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