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# Between exact age 70 and exact age 71 bi di ti life i

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Unformatted text preview: s Binomial model With censoring Example In a mortality investigation, the following data have been recorded for 6 independent lives observed between exact age 70 and exact age 71. bi di ti Life i ai 1 0 1 0 2 0.3 0.9 0 3 0.5 1 1 0.9 4 0 0.4 0 5 0 0.9 1 0.7 6 0 1 1 0.8 Using the Binomial Model of Mortality the likelihood of these observations is LB = p70 × 0.6 p70.3 × 0.5 q70.5 × 0.4 p70 × 0.9 q70 × q70 8/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model The actuarial estimate 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 9/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model The actuarial estimate The actuarial estimate Let D be the number of deaths ( N i =1 Di ). We have then N E [D ] = (bi −ai qx +ai ) i =1 Now note that bi −ai qx +ai = Pr ith life dies between x + ai and x + bi = Pr ith life dies between x + ai and x + 1 − Pr ith life dies between x + bi and x + 1 = 9/30 1−ai qx +ai − (bi −ai px +ai ) (1−bi qx +bi ) Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model The actuarial estimate Furthermore, using Balducci assumption and 1−bi qx +bi = (1 − bi ) qx we obtain 1−ai qx +ai = (1 − ai ) qx N...
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