13_AS_5B_lec_annotated

# Transition rates ij i j from state i to state j is ij

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Unformatted text preview: arkov Model Example (continued) Partial diﬀerentiation: ∂ log L d ∂ log L u = −v + = −w + ∂µ µ ∂ν ν ∂ log L s ∂ log L r = −v + = −w + ∂σ σ ∂ρ ρ Setting these derivatives equal to 0 and solving the equations: d u s r ν= σ= ρ= v w v w Checking the second derivatives (negative), we have maximum likelihood estimates: d u s r µ= ˆ ν= ˆ σ= ˆ ρ= ˆ v w v w Therefore, the maximum likelihood estimators: µ= µ= ˜ 49/60 D V ν= ˜ U S R σ= ˜ ρ= ˜ W V W Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model The maximum likelihood estimates of the transition rates µij (i = j ) from state i to state j is µij = ˆ nij i =j ti Here, nij is the no. of transitions from state i to j ti is the total waiting time in state i . As µii = − j =i µij , the MLE of µii is µii = − ˆ µij ˆ j =i 50/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Statistical properties of estimators For the Able-illness-death model Di − µVi , Ui − ν Wi , Si − σ Vi , Ri − ρWi are uncorrelated (but not necessarily independent) (µ,...
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