13_AS_6_lec_a

n d actuarial statistics module 6 parametric models

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Unformatted text preview: , 1, 2, . . . N . d Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring Maximum Likelihood estimation Under the Binomial model, the loglikelihood is ln L (q ) = ln N d + d ln q + N − d ln (1 − q ) so that d N −d ∂ ln L (q ) = − . ∂q q 1−q First order conditions then yield q= d N Noting ∂2 d (N − d ) ln L (q ) |q= d = − 2 − <0 N ∂ q2 q (1 − q )2 4/30 confirms that q is an MLE. Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model Without censoring Properties of maximum likelihood estimator: unbiased E [q ] = q Var [q ] = efficient) q (1−q ) N (minimum variance of all estimators - − asymptotically q ∼ Normal q , q(1N q) Numerical example: Consider a mortality investigation over 1 year. The population at the beginning of the year numbered 27000 and there were 45 deaths. Estimate q , and also provide a 95% confidence interval. We have q = ˆ 45 27000 5/30 ± 1.96 45 27000 so q (1−q ) ˆ ˆ N that the confidence interval is Actuarial Statistics – Module 6: Parametric models: Binomial and Poisson models Binomial model With censoring 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 6/30 Actuarial Statistics – Mod...
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