Compute the asymptotic standard error of the

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Unformatted text preview: equal to (I (β ))−1 , where I (β ) is the observed information matrix given by ˆ I (β ) = 22/45 − ∂ 2 ln L (β ) |ˆ ∂βi ∂βj β =β i ,j =1,··· ,p Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Estimation of the regression parameters β Example I Given the partial likelihood: L= 1 2 3 23/45 Ce β , (e β + 2)(e β + 3) ˆ1 Show that the MLE of β is β = 2 log 6. Compute the asymptotic standard error of the estimator. Construct an approximate 95% confidence interval for the parameter β . Solution: 1 log L = log C + β − log(e β + 2) − log(e β + 3) log eβ eβ Solving d d β L = 1 − e β +2 − e β +3 = 0 =⇒ β = 2 ˆ Noting that d log L | 1 < 0, β = 1 log 6. 2 dβ β = 2 log 6 2 1 2 log 6 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Estimation of the regression parameters β Example II 2 d 2 log L d β2 β 2e = − (e β +2)2 − 3e β (e β +3)2 2 ˆ I (β ) = [− d dlog L ]β =β The ˆ β2 ˜ asymptotic variance of β is ˆ ˆ [I (β )]−1 = [− 2e β ˆ (e β + Asymptotic standard error 3 − ˆ (e β + ˜ Var (β ) = 3)2 √ −1 = 2.02062 2.02062 = 1.4215 ˜ β is asymptotically normally distributed. A 95% confidence interval for β is ˆ β ±z1−0.05/2 24/45 2)2 ˆ 3e β 1 ˜ var (β ) = log 6±1.96×1.4215 = (...
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This document was uploaded on 04/03/2014.

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