{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

13_AS_2_lec_a

Estimator nelson aalen estimator 4 comparing survival

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: approach Kaplan-Meier (product limit) estimator 1 Introduction 2 Censoring and truncation Censoring Truncation Likelihood for censored and truncated data 3 Estimating the lifetime distribution: non-parametric approach Preliminaries Maximum Likelihood with censoring The likelihood as a function of the discrete hazard function Kaplan-Meier (product limit) estimator Nelson-Aalen estimator 4 Comparing survival functions Introductory example A hypothesis test Special cases 19/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Kaplan-Meier (product limit) estimator Kaplan-Meier estimator Given what we have above, the KM estimator is obvious and given by ∧ ∧ F (t ) = 1 − 1 − λj j :tj ≤t or alternatively ∧ S (t ) = 1 1− j :tj ≤t if t < t1 dj nj t1 ≤ t ≤ tmax . Note the KM estimator is also called Product-limit estimator. 19/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Kaplan-Meier (product limit) estimator 0.0 0.2 0.4 0.6 0.8 1.0 An example of the plot of K-M estimate of a survival function 0 20/42 5 10 15 20 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Kaplan-Meier (product limit) estimator The Kaplan-Meier estimator is well deﬁned for time points less than tmax : if t < t1 1 ∧ ∧ S (t ) if tj ≤ t < tj +1 , j = 1, · · · , k − 1 S (t ) = ∧ j S (tk ) if tk ≤ t < tmax . For estimator of the survival function beyond tmax : If tmax corresponds to a death time and there is no censoring at tmax , the estimated survive curve is ZERO beyond tmax . If tmax = tkck , the value of S (t ) for t > tmax is undetermined. Two extreme views: If assuming that the survivors at time tmax would have died ˆ immediately after tmax , S (t ) = 0 for t > tmax . If assuming that the survivors at time tmax would die at ∞, ˆ ˆ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online