13_AS_2_lec_a

# Of lives at risk n1 10 n2 9 party n3 7 1242 actuarial

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Unformatted text preview: 0 Investigation closes. Re- t3,1 = 17, t3,2 = · · · = t3,5 = 20 maining rats hold street No. of lives at risk n1 = 10, n2 = 9, party. n3 = 7 12/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Maximum Likelihood with censoring 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 13/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Maximum Likelihood with censoring The total likelihood is k F (tj ) − F j =1 tj− dj k cj [1 − F (tjl )] j =0 l =1 This is because (of , and the censored lives surviving to tjl for j = 0, 1, · · · k and l = 1, · · · , cj ) we have dj deaths at time tj for j = 1, 2, · · · , k and their likelihood is F (tj ) − F tj− we have censored lives surviving to tjl for j = 0, 1, · · · k and l = 1, · · · , cj with probability 1 − F (tjl ) 13/42 Note we can take the product thanks to the assumption of non-informative censoring Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach Maximum Likelihood with censoring To maximize the likelihood k F (tj ) − F j =1 tj− dj k cj [1 − F (tjl )] , j =0 l =1 note the following: [1 − F (tjl )] will be maximised if F (tjl ) = F (tj ) i.e. the maximum likelihood estimator F (t ) is constant between tj and tj− (because tj ≤ tjl < tj +1 and F is non-decreasing) +1 F (tj ) > F tj− at each failure, otherwise the likelihood will be zero Therefore, maximum likelihood estimate of F (t ) is a step function with jumps at the times of the observed failures (deaths). This allows us to reformulate the likelihood as follows. 14/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach The likelihood as a function of the discrete hazard function 1 Introduction 2 Censoring and truncation Censoring Truncation Likelihood for censored and truncated data 3 Estimating the lifetime distribution: non-parametric approach Preliminaries...
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