13_AS_4a_lec_a

The weibull is the only continuous distribution that

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Unformatted text preview: ortional hazards: the hazard rate of X with covariates Z is h(x ; Z ) = h0 (x exp(θZ T )) exp(θZ T ), where h0 (x ) is the hazard rate function corresponding to the baseline survival function S0 (x ). The Weibull is the only continuous distribution that yields both a proportional hazards and an accelerated failure-time model. 11/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples Exercise Show that linear log-time representation: Y = log X = µ + β Z T + σ W where W has the standard extreme value distribution is equivalent to accelerated failure-time representation: S (x ; z ) = S0 x exp θz T with S0 (x ) = exp (−λx α ) , where λ and α are some constants. 12/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples Solution S (x ; Z ) = Pr (X > x ) = Pr exp µ + β Z T + σ W > x = Pr W ≥ log x − µ − β Z T σ ∞ = w log x −µ−β Z T σ e e −e de w = −e −e = exp{−e log x −µ−β Z T σ −µ σ } = exp{−e = exp(−λ[x exp(θZ T )]α ) µ 13/15 w w log x −µ−β Z T σ e w −e dw ∞ T w = log x −µ−β Z σ } T log xe −β Z = exp{−e ∞ = 1 where λ = e − σ , α = σ , θ = −β. µ −σ xe −β Z T 1 σ } Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples The log normal regression model Let Y = log X = µ + β z T + σ W . Now assume that W has a standard normal distribution. Note: Y is now lognormal The survival function of X deﬁned above has the accelerated failure-time representation: S (x ; z ) = S0 x exp θz T See proof next. 14/15 Actuarial Statistics – Module 4a: Parametric models: Introduction to more advanced parametric models Examples S (x ; Z ) = Pr (X > x ) = Pr exp µ + β Z T + σ W > x = Pr W ≥ log x − µ − β Z T σ =1−Φ = 1−Φ T log x + log(e −µ ) + log e ((−β )Z ) σ = 1−Φ log x − µ − β Z T σ T log(e −µ ) + log xe ((−β )Z ) σ T T = S0 xe ((−β )Z ) = S0 xe (θZ ) −µ )+log(·) where S0 (·) = 1 − Φ log(e σ , θ = −β , and Φ(·) is the standard normal distribution function. 15/15...
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This document was uploaded on 04/03/2014.

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