3. Let X = {Xk, k = N} be a Markov chain with state-space S = {1, 2,...,m}. The real- izations of the process are observed throughtime k = 0, 1, ..., n, with n < ∞. From the realizations of the sample path, the following observations are recorded n-1 Nij = Σ 1 [X₁ =i‚Xx+1=j} = # (i → j) k=0 m n N₁ = Σ Nij = Σ 1 (X₁ = i)} = #transitions out of state i j=1 k=1 Using these observations, we want to estimate the transition matrix P of the Markov chain. For this purpose, let ik Є S be a state that X occupies in step k, i.e., Xk = ik, k = {0,..., n}. The likelihood function of the realizations (Xo, ..., ✗} of X after n steps reads L= P(X = in, Xn-1 = in-1,..., Xo = io). (a) Let P(Xo = io) = = P(X1 = 1, i.e., X starts in state io with probability one. Let Pij jXoi) be the (i, j)element of transition matrix P. Show that (2) simplifies to = n-1 L =P(X₁ = in|X。 = in−1) × P(X₁ = in−1|X0 = in-2) × ……. P(X₁ = i₁|Xo = io) = [ ] P (b) Given that ik, ik+1 Є S, the likelihood reduces to m m ²-ÛÛ ПП Piy). L = i=1 j=1 Derive the loglikelihood function of the realizations. (c) Show that the maximum likelihood estimate Pij of j Pij is given by k=0 Nij Pij = Ni (d) Show that Pij is a consistent estimator of Pij, i.e., n→∞ Pij Pij. (e) A simulation study of X on state space S = {1, 2, 3} yields the following realization of the sample path $path: 3 1 2 1 3 2 1 2 1 2 1 2 1 2 1 3 1 3 1 2 3 2 3 2 3 2 1 3 1 2 123 13 12 12 Use this realization to estimate the transition matrix P.
3. Let X = {Xk, k = N} be a Markov chain with state-space S = {1, 2,...,m}. The real- izations of the process are observed throughtime k = 0, 1, ..., n, with n < ∞. From the realizations of the sample path, the following observations are recorded n-1 Nij = Σ 1 [X₁ =i‚Xx+1=j} = # (i → j) k=0 m n N₁ = Σ Nij = Σ 1 (X₁ = i)} = #transitions out of state i j=1 k=1 Using these observations, we want to estimate the transition matrix P of the Markov chain. For this purpose, let ik Є S be a state that X occupies in step k, i.e., Xk = ik, k = {0,..., n}. The likelihood function of the realizations (Xo, ..., ✗} of X after n steps reads L= P(X = in, Xn-1 = in-1,..., Xo = io). (a) Let P(Xo = io) = = P(X1 = 1, i.e., X starts in state io with probability one. Let Pij jXoi) be the (i, j)element of transition matrix P. Show that (2) simplifies to = n-1 L =P(X₁ = in|X。 = in−1) × P(X₁ = in−1|X0 = in-2) × ……. P(X₁ = i₁|Xo = io) = [ ] P (b) Given that ik, ik+1 Є S, the likelihood reduces to m m ²-ÛÛ ПП Piy). L = i=1 j=1 Derive the loglikelihood function of the realizations. (c) Show that the maximum likelihood estimate Pij of j Pij is given by k=0 Nij Pij = Ni (d) Show that Pij is a consistent estimator of Pij, i.e., n→∞ Pij Pij. (e) A simulation study of X on state space S = {1, 2, 3} yields the following realization of the sample path $path: 3 1 2 1 3 2 1 2 1 2 1 2 1 2 1 3 1 3 1 2 3 2 3 2 3 2 1 3 1 2 123 13 12 12 Use this realization to estimate the transition matrix P.
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![3. Let X = {Xk, k = N} be a Markov chain with state-space S = {1, 2,...,m}. The real-
izations of the process are observed throughtime k = 0, 1, ..., n, with n < ∞. From the
realizations of the sample path, the following observations are recorded
n-1
Nij = Σ 1 [X₁ =i‚Xx+1=j} = # (i → j)
k=0
m
n
N₁ = Σ Nij = Σ 1 (X₁ = i)} = #transitions out of state i
j=1
k=1
Using these observations, we want to estimate the transition matrix P of the Markov chain.
For this purpose, let ik Є S be a state that X occupies in step k, i.e., Xk = ik, k = {0,..., n}.
The likelihood function of the realizations (Xo, ..., ✗} of X after n steps reads
L= P(X = in, Xn-1 = in-1,..., Xo = io).
(a) Let P(Xo
=
io)
=
=
P(X1
=
1, i.e., X starts in state io with probability one. Let Pij
jXoi) be the (i, j)element of transition matrix P. Show that (2) simplifies to
=
n-1
L =P(X₁ = in|X。 = in−1) × P(X₁ = in−1|X0 = in-2) × ……. P(X₁ = i₁|Xo = io) = [ ] P
(b) Given that ik, ik+1 Є S, the likelihood reduces to
m
m
²-ÛÛ
ПП Piy).
L =
i=1 j=1
Derive the loglikelihood function of the realizations.
(c) Show that the maximum likelihood estimate Pij of j Pij
is given by
k=0
Nij
Pij = Ni
(d) Show that Pij is a consistent estimator of Pij, i.e.,
n→∞
Pij Pij.
(e) A simulation study of X on state space S = {1, 2, 3} yields the following realization
of the sample path
$path: 3 1 2 1 3 2 1 2 1 2 1 2 1 2 1 3 1 3 1 2
3 2 3 2 3 2 1 3 1 2 123 13 12 12
Use this realization to estimate the transition matrix P.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F709bc3b4-4900-4b80-b7ac-ae050f848730%2Fpkso1o7_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let X = {Xk, k = N} be a Markov chain with state-space S = {1, 2,...,m}. The real-
izations of the process are observed throughtime k = 0, 1, ..., n, with n < ∞. From the
realizations of the sample path, the following observations are recorded
n-1
Nij = Σ 1 [X₁ =i‚Xx+1=j} = # (i → j)
k=0
m
n
N₁ = Σ Nij = Σ 1 (X₁ = i)} = #transitions out of state i
j=1
k=1
Using these observations, we want to estimate the transition matrix P of the Markov chain.
For this purpose, let ik Є S be a state that X occupies in step k, i.e., Xk = ik, k = {0,..., n}.
The likelihood function of the realizations (Xo, ..., ✗} of X after n steps reads
L= P(X = in, Xn-1 = in-1,..., Xo = io).
(a) Let P(Xo
=
io)
=
=
P(X1
=
1, i.e., X starts in state io with probability one. Let Pij
jXoi) be the (i, j)element of transition matrix P. Show that (2) simplifies to
=
n-1
L =P(X₁ = in|X。 = in−1) × P(X₁ = in−1|X0 = in-2) × ……. P(X₁ = i₁|Xo = io) = [ ] P
(b) Given that ik, ik+1 Є S, the likelihood reduces to
m
m
²-ÛÛ
ПП Piy).
L =
i=1 j=1
Derive the loglikelihood function of the realizations.
(c) Show that the maximum likelihood estimate Pij of j Pij
is given by
k=0
Nij
Pij = Ni
(d) Show that Pij is a consistent estimator of Pij, i.e.,
n→∞
Pij Pij.
(e) A simulation study of X on state space S = {1, 2, 3} yields the following realization
of the sample path
$path: 3 1 2 1 3 2 1 2 1 2 1 2 1 2 1 3 1 3 1 2
3 2 3 2 3 2 1 3 1 2 123 13 12 12
Use this realization to estimate the transition matrix P.
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