1. Assume that the time-to-event T has Geometric distribution P(T = t) = (1 − λ)¹¹λ, for t = 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 & exit indicator defined respectively by T = min{T, C} and 8 = = 1{T≤C}. We are interested in estimating the parameter 1 from independent random sample (Ti, di), i = 1, 2, . . ., n, of exit time 7 and reason for exit 6 of n independent individuals. Subject Observed ID exit time Exit indicator T δ 1 1 1 2 1 0 3 2 1 4 3 0 5 5 1 Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.) (a) Show that the survival probability S (t) = P(T ≥ t) = (1 − λ)¹¹, for t≥ 1. (b) Show that under independent censoring, the mortality rate under censoring coincides with Mc(t) = P(T = 1,8 = 1|7 ≥ t), µ(t) = P(T = t|T ≥ t) = λ. (c) Write the likelihood function of independent pair observations (T;, ;), i = 1, … … …, n. (d) Find the maximum likelihood estimator λ of the parameter λ. Show your working. Use the dataset in Table 1 to get the value of the estimator λ. (e) Show that the observed Fisher information J(λ) is given by n n 1 J(a): = 22 i=1 1 §i + (1-2)² Σ(; - 1). i=1 (f) Calculate an estimate Var(^) of the variance of λ. Use the dataset. (g) Give the 95% confidence interval for λ.
1. Assume that the time-to-event T has Geometric distribution P(T = t) = (1 − λ)¹¹λ, for t = 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 & exit indicator defined respectively by T = min{T, C} and 8 = = 1{T≤C}. We are interested in estimating the parameter 1 from independent random sample (Ti, di), i = 1, 2, . . ., n, of exit time 7 and reason for exit 6 of n independent individuals. Subject Observed ID exit time Exit indicator T δ 1 1 1 2 1 0 3 2 1 4 3 0 5 5 1 Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.) (a) Show that the survival probability S (t) = P(T ≥ t) = (1 − λ)¹¹, for t≥ 1. (b) Show that under independent censoring, the mortality rate under censoring coincides with Mc(t) = P(T = 1,8 = 1|7 ≥ t), µ(t) = P(T = t|T ≥ t) = λ. (c) Write the likelihood function of independent pair observations (T;, ;), i = 1, … … …, n. (d) Find the maximum likelihood estimator λ of the parameter λ. Show your working. Use the dataset in Table 1 to get the value of the estimator λ. (e) Show that the observed Fisher information J(λ) is given by n n 1 J(a): = 22 i=1 1 §i + (1-2)² Σ(; - 1). i=1 (f) Calculate an estimate Var(^) of the variance of λ. Use the dataset. (g) Give the 95% confidence interval for λ.
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 assist with the following questions with the answers.
Transcribed Image Text:1. Assume that the time-to-event T has Geometric distribution
P(T = t) = (1 − λ)¹¹λ, for t = 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 & exit indicator defined respectively by
T = min{T, C} and 8 = = 1{T≤C}.
We are interested in estimating the parameter 1 from independent random sample (Ti, di),
i = 1, 2, . . ., n, of exit time 7 and reason for exit 6 of n independent individuals.
Subject Observed
ID
exit time
Exit
indicator
T
δ
1
1
1
2
1
0
3
2
1
4
3
0
5
5
1
Table 1: (Exit indicator 8 = 1 if not censored and 8 = 0 if censored.)
(a) Show that the survival probability S (t) = P(T ≥ t) = (1 − λ)¹¹, for t≥ 1.
(b) Show that under independent censoring, the mortality rate under censoring
coincides with
Mc(t) = P(T = 1,8 = 1|7 ≥ t),
µ(t) = P(T = t|T ≥ t) = λ.
(c) Write the likelihood function of independent pair observations (T;, ;), i = 1, … … …, n.
(d) Find the maximum likelihood estimator λ of the parameter λ. Show your working.
Use the dataset in Table 1 to get the value of the estimator λ.
(e) Show that the observed Fisher information J(λ) is given by
n
n
1
J(a):
=
22
i=1
1
§i +
(1-2)²
Σ(; - 1).
i=1
(f) Calculate an estimate Var(^) of the variance of λ. Use the dataset.
(g) Give the 95% confidence interval for λ.
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