13_AS_3_lec_a

# Just prior to time 14 4 lives are at risk and their

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Unformatted text preview: asterisk represent censored observations. Using Breslow’s approximation, what contribution to the partial likelihood is made by deaths at time 14? 20/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Estimation of the regression parameters β Solution Denote β = (β1 , β2 ). Just prior to time 14, 4 lives are at risk and their covariates are (0, 75), (1, 68), (1, 49) and (0, 86). At time 14, there are two deaths with covariates (0, 75), (1, 68). Hence, the contribution to the partial likelihood made by deaths at time 14 is approximated by exp β (0 + 1, 75 + 68)T 2 [exp (β (0, 75)T ) + exp (β (1, 68)T ) + exp (β (1, 49)T ) + exp (β (0, 86)T )] 21/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Estimation of the regression parameters β Properties of the maximum (partial) likelihood estimator The eﬃcient score function is deﬁned by u (β ) = ˆ Solving u β ∂ ln L (β ) ∂ ln L (β ) ,..., ∂β1 ∂βp ˆ = 0 gives maximum likelihood estimate β ˜ The maximum partial likelihood estimator β (of β ) is asymptotically unbiased asymptotically (multivariate) normally distributed with mean ˆ ˆ β and variance (matrix)...
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## This document was uploaded on 04/03/2014.

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