2. Let X = (X,,10) be a continuous-time homogeneous Markov chain with an intensity matrix Q and state space S = [1,2,...,m). The sample path of X is observed continuously on a finite interval [0,T], T X [1] 2 1 3 2 1 2 3 2 1 3 2 1 3 > T [1] 0.00 1.43 7.41 10.71 15.03 17.02 25.12 27.12 32.05 33.00 36.60 44.41 52.08 Based on this realizations, estimate the intensity matrix of X. (d) Find the average time the process spends in each state. Compare your findings against the above realizations.
2. Let X = (X,,10) be a continuous-time homogeneous Markov chain with an intensity matrix Q and state space S = [1,2,...,m). The sample path of X is observed continuously on a finite interval [0,T], T X [1] 2 1 3 2 1 2 3 2 1 3 2 1 3 > T [1] 0.00 1.43 7.41 10.71 15.03 17.02 25.12 27.12 32.05 33.00 36.60 44.41 52.08 Based on this realizations, estimate the intensity matrix of X. (d) Find the average time the process spends in each state. Compare your findings against the above realizations.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![2. Let X = (X,,10) be a continuous-time homogeneous Markov chain with an intensity
matrix Q and state space S = [1,2,...,m). The sample path of X is observed continuously
on a finite interval [0,T], T<o, from which the following statistics are recorded
Nij #i→j transitions of X,
Z; =the amount of time X spends in state i = S in total.
Using these information, we want to estimate the intensity matrix Q. Let qi-qii for
each i = S. It is known that the likelihood function of the sample path is given by
m
m
L(qij i, j = S) = (q) e az
(a) Write down the loglikelihood function.
(b) Show that the maximum likelihood estimator qij of qij is given by
Hint: Recall that q=1 Σji ij
Nij
Jij Zi
(c) In a numerical study, a sample path is observed within time window [0,50]. Con-
sider the following realization of sample paths
> X
[1] 2 1 3 2 1 2 3 2 1 3 2 1 3
> T
[1] 0.00 1.43 7.41 10.71 15.03 17.02 25.12 27.12
32.05 33.00 36.60 44.41 52.08
Based on this realizations, estimate the intensity matrix of X.
(d) Find the average time the process spends in each state. Compare your findings
against the above realizations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F8dc63f6e-ffbe-46c1-9f1f-3d4d62fc401a%2Fvyohw2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let X = (X,,10) be a continuous-time homogeneous Markov chain with an intensity
matrix Q and state space S = [1,2,...,m). The sample path of X is observed continuously
on a finite interval [0,T], T<o, from which the following statistics are recorded
Nij #i→j transitions of X,
Z; =the amount of time X spends in state i = S in total.
Using these information, we want to estimate the intensity matrix Q. Let qi-qii for
each i = S. It is known that the likelihood function of the sample path is given by
m
m
L(qij i, j = S) = (q) e az
(a) Write down the loglikelihood function.
(b) Show that the maximum likelihood estimator qij of qij is given by
Hint: Recall that q=1 Σji ij
Nij
Jij Zi
(c) In a numerical study, a sample path is observed within time window [0,50]. Con-
sider the following realization of sample paths
> X
[1] 2 1 3 2 1 2 3 2 1 3 2 1 3
> T
[1] 0.00 1.43 7.41 10.71 15.03 17.02 25.12 27.12
32.05 33.00 36.60 44.41 52.08
Based on this realizations, estimate the intensity matrix of X.
(d) Find the average time the process spends in each state. Compare your findings
against the above realizations.
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