By Torben Martinussen
This ebook experiences and applies smooth versatile regression versions for survival information with a different specialize in extensions of the Cox version and replacement types with the purpose of describing time-varying results of explanatory variables. Use of the steered versions and techniques is illustrated on actual information examples, utilizing the R-package timereg constructed through the authors, that's utilized in the course of the ebook with labored examples for the knowledge units.
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An important simple consequence of the theorem is that, if the cadlag adapted process X is a local submartingale, then it has a compensator since the constant process 0 is a local submartingale. ˜ be local square integrable martingales. By Jensen’s inLet M and M 2 equality, M is a local submartingale since E(M 2 (t) | Fs ) ≥ (E(M (t) | Fs ))2 = M 2 (s) and hence, by the Doob-Meyer decomposition, it has a compensator. This compensator is denoted by M, M , or more compactly M , and is termed ˜ = 1 (M + M ˜ )2 − the predictable variation process of M .
1) with θ being some parameter of interest. 1). As argued above, N ∗ will typically not be fully observable, but only an incomplete version will be available. The observable part of Ni∗ (t) may often be expressed as t Ni (t) = 0 Ci (s) dNi∗ (s), where Ci (t) denotes the ith so-called filtering process Ci (t) = I(t ∈ Ai ). We require, for simplicity, that the filtering processes are independent across subjects. The principal example of filtering is right-censoring, where Ai = [0, Ui ] with Ui some random censoring time, that is, Ci (t) = I(t ≤ Ui ).
Let T = T ∗ ∧ C, ∆ = I(T ∗ ≤ C) and N (t) = I(T ≤ t, ∆ = 1). 29) to show that the intensity of N (t) with respect to the history σ{I(T ≤ s, ∆ = 0), I(T ≤ s, ∆ = 1) : s ≤ t} is λ(t) = I(t ≤ T )α(t). 1. 28). (b) From the same example, show that: sup |n1/2 s∈[0,t] s 0 P (J(u) − 1)α(u) du| → 0. 9 (Asymptotic results for the Nelson-Aalen estimator) Let N (n) (t) be a counting process satisfying the multiplicative intensity structure λ(t) = Y (n) (t)α(t) with α(t) being locally integrable. The Nelson-Aalen estimator t of 0 α(s) ds is 1 Aˆ(n) (t) = dN (n) (s).