Linear log time model log x z t w dene x0 exp

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Unformatted text preview: accelerated failure-time models Consider the linear log-time model: log X = µ + β Z T + σ W . Define X0 = exp(µ + σ W ) and let S0 (x ) be the survival function of X0 . Then the survival function of X : S (x ; Z ) = Pr (X > x ) = Pr exp µ + β Z T + σ W > x = Pr exp [µ + σ W ] > x exp −β Z T = Pr X0 > x exp −β Z T = S0 x exp −β Z T Hence, the linear log-time model is equivalent to the accelerated failure-time model with θ = −β and S0 (x ) = Pr (exp(µ + σ W ) > x ). 6/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples 1 Linear log-time model 2 Accelerated failure-time model 3 Relationship between the log-time and accelerated failure-time models 4 Examples 7/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples A variety of distributions can be used for the lifetime distribution, some common distributions: Exponential Weibull Log normal Log logistic ··· See K&M, Chapter 12. 7/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples Weibull Distribution The Weibull distribution with scale parameter λ, λ > 0, shape parameter α, α > 0, x > 0 has a hazard rate of µ (x ) = λαx (α−1) This is very flexible. It allows: increasing hazard rates if α > 1, decreasing hazard rates if α < 1, constant hazard rates if α = 1 8/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples K&M, page 23 9/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples Accelerated failure-time representation: S (x ; z ) = S0 x exp θZ T where the baseline survival function S0 (x ) = exp (−λx α ) . Equivalent linear log-time representation: Y = log X = µ + β Z T + σ W where W has the standard extreme value distribution exp(w − e w ), 10/15 −∞ < w < ∞. Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples Proportionality The model has prop...
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This document was uploaded on 04/03/2014.

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