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 .

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter2: Exponential, Logarithmic, And Trigonometric Functions

Section2.CR: Chapter 2 Review

Problem 111CR: Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data...

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