Ex ex ex e e furthermore since is an mle it

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Unformatted text preview: 70 we have ˆ 22/30 q70 = 1 − e −µ70 = 1 − e −0.769 = 0.537 ˆ Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Poisson model Maximum Likelihood Estimation Properties of the estimator c Since D has a Poisson(µEx ) distribution, c E [D ] = Var [D ] = µEx and we have µE c D = cx , and c Ex Ex µE c µ D Var [˜] = Var [ c ] = c x 2 = c . µ Ex ( Ex ) Ex E [˜] = E [ µ Furthermore, since µ is an MLE it is asymptotically normally ˜ distributed: µ µ ∼ N µ, c ˜ Ex 23/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Poisson model Maximum Likelihood Estimation Example In a mortality investigation covering a 5-year period, where the force of mortality can be assumed to be constant, there were 46 deaths and the population remained approximately constant 7,500. The MLE is then µ= µ Since µ ∼ N µ, E c ˜ x µ: µ ± 1.96 24/30 46 d = = 0.00123 c Ex 7, 500 × 5 we can obtain a 95% confidence interval for µ ˆ = 0.00123 ± 1.96 c Ex 0.00123 = (0.00087, 0.00158). 7, 500 × 5 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Poisson model Link to 2-state Markov models 1 Binomial model Without censoring With censoring The a...
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