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13_AS_5B_lec_annotated

4660 actuarial statistics module 5 parametric methods

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Unformatted text preview: ikelihood estimators In a time-homogeneous Markov jump process with state space {1, 2, · · · , m}, we observe Total number of samples: N (l ) (l ) Ti = N Ti : total waiting time in state i , where Ti l =1 the waiting time in state i of l th life. Nij = (l ) N i =1 Nij : is the total number of transitions from state i (l ) to state j , where Nij is the number of transitions from state i to state j made by l th life. 46/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Maximum likelihood estimators As usual, we use lower case symbols for the observed samples For life l , the likelihood is m i =1 n (l ) e −µi ti j =i (l ) µijij The total likelihood is (by multiplying over all l ) : N l =1 m i =1 47/60 m i =1 e −µi ti (l ) e −µi ti n j =i µijij n (l ) ij j =i µij = Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Example (continued) As usual, we use lower case symbols for the observed samples. For life i , the likelihood is proportional to e −(µ+σ)vi e −(ν +ρ)wi µdi ν ui σ si ρri The total likelihood: L (σ, ρ, µ, ν ) N e −(µ+σ)vi e −(ν +ρ)wi µdi ν ui σ si ρri = i =1 −(µ+σ )v −(ν +ρ)w d u s r =e = e e −µv d µ µνσρ e −σ v σs e −ν w ν u e −ρw ρr So the log-likelihood: log L = −(µ + σ )v − (ν + ρ)w + d log µ + u log µ + s log σ + r log ρ 48/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model General M...
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