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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.

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.

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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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.
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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