13_AS_6_lec_a

The mean and variance are available exactly in terms

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Unformatted text preview: s no difference between the two-state and Poisson models, because the maximum likelihood estimates are the same 27/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Discussion: Binomial, Poisson and Markov multi-state models Statistical properties I In the Markov model - MLE is consistent, asymptotically unbiased and normally distributed; the variance is only available asymptotically In the Poisson model, the MLE is consistent and unbiased. The mean and variance are available exactly in terms of the true µ, but are estimated form the data by the same expressions as estimate the asymptotic mean and variance in the two-state Markov model. 28/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Discussion: Binomial, Poisson and Markov multi-state models Statistical properties II In the “naive” Binomial model, the MLE is consistent and unbiased and the exact mean and variance can be expressed in terms of the true qx . However, in practice, the data can rarely conform to the “naive” model, so only approximate results are available. When µ is small, none of these models are better than the other models on the basis of the statistical properties of the MLEs alone. 29/30 Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Discussion: Binomial, Poisson and Markov multi-state models More general applications Markov multiple state model extends easily to many decrements and more complicated models Poisson model extends easily to multiple decrements, but not to processes with increments (immigration, new entrants) Binomial model does not extend easily to multiple decrements 30/30...
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