13_AS_2_lec_a

Or integrated hazard t t s ds 0 mj j tj t since

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Unformatted text preview: 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 26/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Nelson-Aalen estimator ˆ Another non-parametric approach for F (t ) is the Nelson-Aalen estimator (from Nelson, 1971 and Aalen, 1978), which is based on the cumulative hazard (or integrated hazard) t Λt = µs ds + 0 mj . j :tj ≤t ˆ Since in our setting we do not have continuous increases in F (t ) (only jumps – see previous sub-section) we focus on the second half only and use the ML estimator for λj to approximate the mj ’s such that dj ˆ Λt = nj j :tj ≤t Finally, ˆ ˆ S ( t ) = e − Λt . 26/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Nelson-Aalen estimator Variance of Nelson-Aalen estimator ˜ Let Λ (t ) denote the estimatOR. Its variance is approximated as (p. 33 of yellow book): ˜ var Λ (t ) ≈ j :tj ≤t dj (nj − dj ) nj3 Example Consider the following recorded data for the lifetime of a group of rats 2, 4+ , 5, 8, 12, 12+ , 17, 18+ , 18+ , 18+ , where + denotes the censored. Derive 95% confidence intervals for ˆ the N-A estimators Λ(5). 27/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Nelson-Aalen estimator Relationship between the Kaplan-Meier and Nelson-Aalen Denote: ˆ the Kaplan-Meier estimate of the survival function by SKM (t ) ˆ the Nelson-Aslen estimate of the survival function by SNA (t ) Then ˆ SKM (t ) = 1− j :tj ≤t dj nj ≈ exp − j :tj ≤t (using e −x ≈ 1 − x for small |x |) ∧ dj = exp −Λt nj ˆ = SNA (t ) Furthermore, ˆ ˆ SKM (t ) < SNA (t ) 28/42 for t1 ≤ t ≤ tmax . (remember ln(1 − x ) < −x for 0 < x < 1) Actuarial Statistics – Module 2: Non-parametric methods Comparing survival functions 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 fu...
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