13_AS_6_lec_a

# Deaths d is poisson if they can die several times c

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Unformatted text preview: period for each individual in age interval (x , x + 1) c The sum of all of those observed periods is Ex , or observed waiting time v This means that the time to death of each individual (within that year) is exponential, and hence (iﬀ) that the number of deaths D is Poisson (if they can die several times): c Pr [D = d ] = c e −µEx (µEx )d , d! d = 0, 1, 2, . . . c with E [D ] = Var (D ) = µEx . Note that this implicitly allows for the possibility that there will be more than N deaths. 19/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Poisson model Maximum Likelihood Estimation 1 Binomial model Without censoring With censoring The actuarial estimate Central exposed to risk Discussion 2 Poisson model Introduction Maximum Likelihood Estimation Link to 2-state Markov models 3 Discussion: Binomial, Poisson and Markov multi-state models 20/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Poisson model Maximum Likelihood Estimation Maximum Likelihood Estimation Likelihood of observing d deaths is c c e −µEx (µEx )d L ( µ) = d! with Log-likelihood c c ln L (µ) = −µEx + d (ln µ + ln Ex ) − ln (d !) . Diﬀerentiate 20/30 ∂ d c ln L (µ) = − Ex ∂µ µ ∂2 d ln L (µ) = − 2 < 0 2...
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