1. Assume that individual lifetime T follows a Geometric distribution: P(T = x) = (1 − A)¹-¹a, for _x=1,2,..., with 0 << 1. We are interested in estimating the survival probability S(x) := P(T ≥ x) at age of x and the intensity from a given n independent observations T₁,...,Tn of T. (a) Show that the mortality rate (x) = P(T = x|T ≥ x) does not depend on the age, i.e., λ(x) = λ for all x. (b) Write down the likelihood and log-likelihood function of the observations {T}. (c) Show that the maximum likelihood estimation A, is given by An J(X) = (d) Show that is a consistent estimator of λ. (e) Show that the observed Fisher Information J() is given by n 2² + (1 n 71 Σ Ti 1 (T₁ - 1). (f) Find an estimate of the variance Var(n). (g) Use the following sample dataset of T to answer the questions below. 13, 15, 8, 6, 5, 25, 7, 8, 35, 41 i. Calculate , and give a 95% confidence interval of 1. ii. Give an estimate¹ S(x) of the survival probability of individual at the age of 45.
1. Assume that individual lifetime T follows a Geometric distribution: P(T = x) = (1 − A)¹-¹a, for _x=1,2,..., with 0 << 1. We are interested in estimating the survival probability S(x) := P(T ≥ x) at age of x and the intensity from a given n independent observations T₁,...,Tn of T. (a) Show that the mortality rate (x) = P(T = x|T ≥ x) does not depend on the age, i.e., λ(x) = λ for all x. (b) Write down the likelihood and log-likelihood function of the observations {T}. (c) Show that the maximum likelihood estimation A, is given by An J(X) = (d) Show that is a consistent estimator of λ. (e) Show that the observed Fisher Information J() is given by n 2² + (1 n 71 Σ Ti 1 (T₁ - 1). (f) Find an estimate of the variance Var(n). (g) Use the following sample dataset of T to answer the questions below. 13, 15, 8, 6, 5, 25, 7, 8, 35, 41 i. Calculate , and give a 95% confidence interval of 1. ii. Give an estimate¹ S(x) of the survival probability of individual at the age of 45.
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 the following questions with full handwritten working out
![1. Assume that individual lifetime T follows a Geometric distribution:
P(T = x) = (1 - A)¹¹, for x= 1,2,...,
with 0 << 1. We are interested in estimating the survival probability S(x) := P(T ≥ x)
at age of x and the intensity from a given n independent observations T₁,...,Tn of T.
(a) Show that the mortality rate (x) = P(T = x|T ≥ x) does not depend on the age, i.e.,
λ(x) = λ for all x.
(b) Write down the likelihood and log-likelihood function of the observations {T}.
(c) Show that the maximum likelihood estimation A, is given by
An
J(X) =
(d) Show that is a consistent estimator of λ.
(e) Show that the observed Fisher Information J() is given by
n
2²
+
(1
n
71
Σ Ti
1
(T₁ - 1).
(f) Find an estimate of the variance Var(n).
(g) Use the following sample dataset of T to answer the questions below.
13, 15, 8, 6, 5, 25, 7, 8, 35, 41
i. Calculate , and give a 95% confidence interval of .
ii. Give an estimate¹ S(x) of the survival probability of individual at the age of 45.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2Ff10b5c21-1b7b-4559-906e-c7a79b5e4a28%2Fw66axm_processed.png&w=3840&q=75)
Transcribed Image Text:1. Assume that individual lifetime T follows a Geometric distribution:
P(T = x) = (1 - A)¹¹, for x= 1,2,...,
with 0 << 1. We are interested in estimating the survival probability S(x) := P(T ≥ x)
at age of x and the intensity from a given n independent observations T₁,...,Tn of T.
(a) Show that the mortality rate (x) = P(T = x|T ≥ x) does not depend on the age, i.e.,
λ(x) = λ for all x.
(b) Write down the likelihood and log-likelihood function of the observations {T}.
(c) Show that the maximum likelihood estimation A, is given by
An
J(X) =
(d) Show that is a consistent estimator of λ.
(e) Show that the observed Fisher Information J() is given by
n
2²
+
(1
n
71
Σ Ti
1
(T₁ - 1).
(f) Find an estimate of the variance Var(n).
(g) Use the following sample dataset of T to answer the questions below.
13, 15, 8, 6, 5, 25, 7, 8, 35, 41
i. Calculate , and give a 95% confidence interval of .
ii. Give an estimate¹ S(x) of the survival probability of individual at the age of 45.
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