2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1-A)-¹, for x = 1,2,..., with 0 << 1. It is subject to censoring which occurs at random time C, independent of T. Denote by r the observed exit time and by 8 exit indicator defined respectively by 7 = min{T,C) and 8 = 1(TC). We are interested in estimating the parameter from independent random sample (T₁, 8;), i = 1,2,..., n, of exit time r and reason for exit & of n independent individuals. Subject ID coincides with Observed exit time T 1 2 4 3 5 5 Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.) Exit indicator (a) Show that under independent censoring, the mortality rate under censoring Ha(x) = P(T = x, 6 = 1|T ≥ x), 8 1 0 1 0 μ(x) = P(T=x|T > x) = λ. J(X) = (b) Write the likelihood function of independent pair observations (T₁, 6;), i = 1, ..., n. (c) Find the maximum likelihood estimator of the parameter A. Show your working. Use the dataset in Table 1 to get the value of the estimator A. (d) Show that the observed Fisher information J() is given by Σ 97² = D + 19 3 4 (T₁-1). (e) Calculate an estimate Var() of the variance of A. Use the dataset. (f) Give the 95% confidence interval for .
2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1-A)-¹, for x = 1,2,..., with 0 << 1. It is subject to censoring which occurs at random time C, independent of T. Denote by r the observed exit time and by 8 exit indicator defined respectively by 7 = min{T,C) and 8 = 1(TC). We are interested in estimating the parameter from independent random sample (T₁, 8;), i = 1,2,..., n, of exit time r and reason for exit & of n independent individuals. Subject ID coincides with Observed exit time T 1 2 4 3 5 5 Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.) Exit indicator (a) Show that under independent censoring, the mortality rate under censoring Ha(x) = P(T = x, 6 = 1|T ≥ x), 8 1 0 1 0 μ(x) = P(T=x|T > x) = λ. J(X) = (b) Write the likelihood function of independent pair observations (T₁, 6;), i = 1, ..., n. (c) Find the maximum likelihood estimator of the parameter A. Show your working. Use the dataset in Table 1 to get the value of the estimator A. (d) Show that the observed Fisher information J() is given by Σ 97² = D + 19 3 4 (T₁-1). (e) Calculate an estimate Var() of the variance of A. Use the dataset. (f) Give the 95% confidence interval for .
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please do question 2e and 2f with full handwritten working out
Transcribed Image Text:2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution
P(T = x) = (1-A)-¹, for x = 1,2,...,
with 0 << 1. It is subject to censoring which occurs at random time C, independent of
T. Denote by r the observed exit time and by 8 exit indicator defined respectively by
7 = min{T,C) and 8 = 1(TC).
We are interested in estimating the parameter from independent random sample (T₁, 8;),
i = 1,2,..., n, of exit time r and reason for exit & of n independent individuals.
Subject
ID
coincides with
Observed
exit time
T
1
2
4
3
5
5
Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.)
Exit
indicator
(a) Show that under independent censoring, the mortality rate under censoring
Ha(x) = P(T = x, 6 = 1|T ≥ x),
8
1
0
1
0
μ(x) = P(T=x|T > x) = λ.
J(X) =
(b) Write the likelihood function of independent pair observations (T₁, 6;), i = 1, ..., n.
(c) Find the maximum likelihood estimator of the parameter A. Show your working.
Use the dataset in Table 1 to get the value of the estimator A.
(d) Show that the observed Fisher information J() is given by
Σ
97² = D + 19 3 4
(T₁-1).
(e) Calculate an estimate Var() of the variance of A. Use the dataset.
(f) Give the 95% confidence interval for .
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