13_AS_2_lec_a

Comparing survival functions introductory example a

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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 15/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach The likelihood as a function of the discrete hazard function Discrete hazard function Suppose F (t ) corresponds has positive probability masses at and only at the points t1 < t2 < . . . < tk (we’ve just shown this had to be the case). Deﬁne the discrete hazard function as λj = Pr [T = tj |T ≥ tj ] , (1 ≤ j ≤ k ) . Then S (t ) = 1 − F (t ) = (1 − λj ) j :tj ≤t 15/42 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach The likelihood as a function of the discrete hazard function Proof Note that since t0 = 0 we have P (T > t0 ) = P (T > 0) = 1 Then P (T > t1 ) = P (T = t1 ) P (T > t1 ) =1− = 1 − λ1 P (T > t0 ) P (T > t0 ) More generally 1 − λj = P (T > tj ) P (T > tj ) = P (T ≥ tj ) P (T > tj −1 ) and therefore (by induction) S (t ) = P (T > t ) = j :tj ≤t 16/42 P (T > tj ) = P (T > tj −1 ) (1 − λj ) j :tj ≤t Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach The likelihood as a function of the discrete hazard function Now let us rewrite the total likelihood as dj k k F (tj ) − F tj− 1 − F tj− L= − 1 − F tj j =0 j =1 dj cj [1 − F (tjl )] l =1 where F (0) = 0 and d0 = 0. This can be simpliﬁed since j −1 1 − F (tj− ) = 1 − F (tj −1 ) = S (tj −1 ) = (1 − λi ) = i :ti ≤tj −1 (1 − λi ) i =1 and j 1 − F (tjl ) = 1 − F (tj ) = S (tj ) = (1 − λi ) = i :ti ≤tj 17/42 (1 − λi ) i =1 Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric approach The likelihood as a function of the discrete hazard function Now note that F (tj ) − F tj− 1 − F tj− = λj and that nj = dj + cj + nj +1 forj = 1, · · · , k Hence, k [λj ]dj [1 − λj ]nj −dj L= j =1 which is in ‘binomial’ form and has maximum likelihood estimator ∧ λj = 18/42 dj nj (1 ≤ j ≤ k ) Actuarial Statistics – Module 2: Non-parametric methods Estimating the lifetime distribution: non-parametric...
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