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j) = ¡j. Suppose that sample paths of Z are observed from which we record N₁ =#i → j transitions of Z, N₁ = N₁, (# transitions out of state i). s known that the likelihood function of the observations of Z is given by 4-000 L= orporating the constraint Σ= 1 for each i ES, Write the loglikelihood function of the observation. ) Show that the maximum likelihood estimator of i→ j transition probability is Nu ) Let m = 2 and assume that the number of transitions of Z is given by the matrix 50 100 Use the above information to derive estimators for 11, 12, 21 and 22-

j) = ¡j. Suppose that sample paths of Z are observed from which we record N₁ =#i → j transitions of Z, N₁ = N₁, (# transitions out of state i). s known that the likelihood function of the observations of Z is given by 4-000 L= orporating the constraint Σ= 1 for each i ES, Write the loglikelihood function of the observation. ) Show that the maximum likelihood estimator of i→ j transition probability is Nu ) Let m = 2 and assume that the number of transitions of Z is given by the matrix 50 100 Use the above information to derive estimators for 11, 12, 21 and 22-

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2. Let Zbe a Markov chain on a state space S = {1,… … , m} with transition probability matrix P(i, j) = Tij. Suppose that sample paths of Z are observed from which we record Njj #i →→ j transitions of Z, m N₁ = [N₁; (# transitions out of state i). j=1 It is known that the likelihood function of the observations of Z is given by m Nij ΠΠΠ. i=1_ j=1 L = Incorporating the constraint j = 1 for each i = S, j=1 m (a) Write the loglikelihood function of the observation. (b) Show that the maximum likelihood estimator of i → j transition probability ij is N = Tij = (c) Let m = 2 and assume that the number of transitions of Z is given by the matrix Nij N₁ 50 100 25 75 Use the above information to derive estimators for 11, 12, 721 and 22.
Transcribed Image Text:2. Let Zbe a Markov chain on a state space S = {1,… … , m} with transition probability matrix P(i, j) = Tij. Suppose that sample paths of Z are observed from which we record Njj #i →→ j transitions of Z, m N₁ = [N₁; (# transitions out of state i). j=1 It is known that the likelihood function of the observations of Z is given by m Nij ΠΠΠ. i=1_ j=1 L = Incorporating the constraint j = 1 for each i = S, j=1 m (a) Write the loglikelihood function of the observation. (b) Show that the maximum likelihood estimator of i → j transition probability ij is N = Tij = (c) Let m = 2 and assume that the number of transitions of Z is given by the matrix Nij N₁ 50 100 25 75 Use the above information to derive estimators for 11, 12, 721 and 22.
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