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...
Related questions
Question
Please do question 2e and 2f with full handwritten working out
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage