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2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1 - A)-¹1, 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(rc). We are interested in estimating the parameter λ from independent random sample (T;, 8;), i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals. Subject ID 2 3 4 coincides with Observed exit time 1 1 2 3 Exit indicator 8 1 0 1 0 Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.) (a) Show that under independent censoring, the mortality rate under censoring H₂(x) = P(r = x, 6 = 1|T 2 x), μ(x) = P(T=x|T ≥ x) = 1. (b) Write the likelihood function of independent pair observations (T¡, di), 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. (d) Show that the observed Fisher information J() is given by

2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1 - A)-¹1, 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(rc). We are interested in estimating the parameter λ from independent random sample (T;, 8;), i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals. Subject ID 2 3 4 coincides with Observed exit time 1 1 2 3 Exit indicator 8 1 0 1 0 Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.) (a) Show that under independent censoring, the mortality rate under censoring H₂(x) = P(r = x, 6 = 1|T 2 x), μ(x) = P(T=x|T ≥ x) = 1. (b) Write the likelihood function of independent pair observations (T¡, di), 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. (d) Show that the observed Fisher information J() is given by

MATLAB: An Introduction with Applications
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 2d, 2e and 2f with full handwritten working out
2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1-2)-¹1, 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 À from independent random sample (T;, 8;), i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals. Subject ID 2 3 4 coincides with Observed exit time 1 1 2 3 Exit indicator Table 1: (Exit indicator d = 1 if not censored and 8 = 0 if censored.) 8 1 0 1 0 (a) Show that under independent censoring, the mortality rate under censoring Ha(x) = P(T= x,8 = 1|T ≥ x), J(X)= μ(x) = P(T=x|T ≥ x) = λ (b) Write the likelihood function of independent pair observations (T¡, d¡), 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 . (d) Show that the observed Fisher information J() is given by — Σ* + ²² ²³ ² di (1- (T₁-1). (e) Calculate an estimate Var() of the variance of λ. Use the dataset. (f) Give the 95% confidence interval for 1.
Transcribed Image Text:2. (Censored lifetime) Assume that the lifetime time T has Geometric distribution P(T = x) = (1-2)-¹1, 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 À from independent random sample (T;, 8;), i=1,2,..., n, of exit time 7 and reason for exit 8 of n independent individuals. Subject ID 2 3 4 coincides with Observed exit time 1 1 2 3 Exit indicator Table 1: (Exit indicator d = 1 if not censored and 8 = 0 if censored.) 8 1 0 1 0 (a) Show that under independent censoring, the mortality rate under censoring Ha(x) = P(T= x,8 = 1|T ≥ x), J(X)= μ(x) = P(T=x|T ≥ x) = λ (b) Write the likelihood function of independent pair observations (T¡, d¡), 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 . (d) Show that the observed Fisher information J() is given by — Σ* + ²² ²³ ² di (1- (T₁-1). (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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