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2. (a) A Markov chain X = {X, 0} is used for a competing risk analysis based on the following simple diagram with the respective transition rates H E1 7 5 E2 Figure 1: Transition diagram of a simple competing risk model. where H represents a healthy state, E₁ event of interest, and E2 other type of exit. i. Write down the intensity matrix of the Markov chain. ii. Find the transition probability Phe₁(t) = P(X₁ = E₁|X0 = H) of moving from a healthy state to the event of interest. iii. Find the expected remaining lifetime of a healthy subject. = (b) From the observations of sample paths of Markov chain X {X, t≥ 0} with state space S = {1,2,3}, the number of transitions Nij between states i, j = S and the length of time Z; spent in each state i = S are reported in the matrices below 0 20 30 10 N = 40 0 50 10 20 0 Z = 20 5 Elements qij of the intensity matrix Q is found as the maximizer of likelihood function 3 3 L({ij}) = [[((ij) i=1 j#i i. Find the estimate 7i; of the intensity qij. ii. Use the given observations to find an estimate of the intensity matrix Q.

2. (a) A Markov chain X = {X, 0} is used for a competing risk analysis based on the following simple diagram with the respective transition rates H E1 7 5 E2 Figure 1: Transition diagram of a simple competing risk model. where H represents a healthy state, E₁ event of interest, and E2 other type of exit. i. Write down the intensity matrix of the Markov chain. ii. Find the transition probability Phe₁(t) = P(X₁ = E₁|X0 = H) of moving from a healthy state to the event of interest. iii. Find the expected remaining lifetime of a healthy subject. = (b) From the observations of sample paths of Markov chain X {X, t≥ 0} with state space S = {1,2,3}, the number of transitions Nij between states i, j = S and the length of time Z; spent in each state i = S are reported in the matrices below 0 20 30 10 N = 40 0 50 10 20 0 Z = 20 5 Elements qij of the intensity matrix Q is found as the maximizer of likelihood function 3 3 L({ij}) = [[((ij) i=1 j#i i. Find the estimate 7i; of the intensity qij. ii. Use the given observations to find an estimate of the intensity matrix Q.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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2. (a) A Markov chain X = {X, 0} is used for a competing risk analysis based on the following simple diagram with the respective transition rates H E1 7 5 E2 Figure 1: Transition diagram of a simple competing risk model. where H represents a healthy state, E₁ event of interest, and E2 other type of exit. i. Write down the intensity matrix of the Markov chain. ii. Find the transition probability Phe₁(t) = P(X₁ = E₁|X0 = H) of moving from a healthy state to the event of interest. iii. Find the expected remaining lifetime of a healthy subject. = (b) From the observations of sample paths of Markov chain X {X, t≥ 0} with state space S = {1,2,3}, the number of transitions Nij between states i, j = S and the length of time Z; spent in each state i = S are reported in the matrices below 0 20 30 10 N = 40 0 50 10 20 0 Z = 20 5 Elements qij of the intensity matrix Q is found as the maximizer of likelihood function 3 3 L({ij}) = [[((ij) i=1 j#i i. Find the estimate 7i; of the intensity qij. ii. Use the given observations to find an estimate of the intensity matrix Q.
Transcribed Image Text:2. (a) A Markov chain X = {X, 0} is used for a competing risk analysis based on the following simple diagram with the respective transition rates H E1 7 5 E2 Figure 1: Transition diagram of a simple competing risk model. where H represents a healthy state, E₁ event of interest, and E2 other type of exit. i. Write down the intensity matrix of the Markov chain. ii. Find the transition probability Phe₁(t) = P(X₁ = E₁|X0 = H) of moving from a healthy state to the event of interest. iii. Find the expected remaining lifetime of a healthy subject. = (b) From the observations of sample paths of Markov chain X {X, t≥ 0} with state space S = {1,2,3}, the number of transitions Nij between states i, j = S and the length of time Z; spent in each state i = S are reported in the matrices below 0 20 30 10 N = 40 0 50 10 20 0 Z = 20 5 Elements qij of the intensity matrix Q is found as the maximizer of likelihood function 3 3 L({ij}) = [[((ij) i=1 j#i i. Find the estimate 7i; of the intensity qij. ii. Use the given observations to find an estimate of the intensity matrix Q.
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