To check whether ej behaves as a sample from a unit

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Unformatted text preview: censored sample from a unit exponential distribution. 37/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Diagnostics for the Cox regression model Proof 1 − S (Tj ; Zj ) ∼ Uniform (0,1) So S (Tj ; Zj ) ∼ Uniform (0,1) Define Ej = − log(S (Tj ; Zj )). Then Ej ∼ Exponential(1) Define C j = − log(S (Cj ; Zj )). Notice that − log(S (Xj ; Zj )) = − log(S (Tj ; Zj )) if Tj ≤ Cj i .e . Ej ≤ C j − log(S (Cj ; Zj )) if Tj > Cj i .e . Ej > C j Consider Ej as a new lifetime r.v. and C j the corresponding censoring time. Then − log(S (Xj ; Zj )) is just a right censored, exponentially distributed version of Ej with censoring time C j . 38/45 Actuarial Statistics – Module 3: Semi-parametric methods: Cox Regression Model Diagnostics for the Cox regression model To check the goodness of fit of a Cox regression model, we need to check wether the Cox-Snell residuals: p ˆ ˆ ej = − log(S (Xj ; zj )) = Λ0 (Xj ) exp( T bk zjk ) k =1 behave as samples from a unit expone...
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