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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 64E

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