13_AS_3_lec_a

1445 actuarial statistics module 3 semi parametric

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Unformatted text preview: ome observations are censored at an observed lifetime. This can significantly complicate the partial likelihood as one needs to include the lives censored at time tj in the risk set R (tj ) and all permutations of simultaneous events. 14/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Estimation of the regression parameters β Case 1 - one death at each observed lifetime The partial likelihood for the death at tj is the first death of the lives Pr the individual (j ) dies at tj in R (tj ) occurs at tj and only one death at tj = i 15/45 λ tj ; Z(j ) = λ (tj ; Zi ) R (tj ) λ0 (tj ) exp β Z(T) j λ0 (tj ) exp β ZiT i R (tj ) = exp β Z(T) j exp β ZiT i R (tj ) Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Estimation of the regression parameters β Multiplying the likelihood over all deaths gives the partial likelihood, whose maximum is our estimate for β : T k exp β Z(j ) ˆ = β L (β ) = is maximal β exp β ZiT j =1 i R (t...
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